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Are Java getters and setters simply methods?

Is the follwing true: Java getters and setters are just methods.
I know they are implemented to manipulate or get the value of private ... attributes/fields. But is it okay to call them methods?

Yes it's okay they are just methods!

Technically, from the language and VM point of view, yes, they are just methods.
Some libraries and frameworks, however recognize their special nature. For example, JPA may be used to map “properties” by annotating either fields or methods. Beans Binding library uses them to access properties, so if you have a getText() / setText() pair on some object, then you can bind the “text” property to some other property of some other object (you'll still have to do the addPropertyChangeListener magic, though). But this is just a “convention over configuration” phenomena combined with the power of Reflection.

As per JLS,
A method declares executable code that can be invoked, passing a fixed number of values as arguments.
And this criterias satisfies to getters and setters as well; so we can say the are "methods" in java language.

Luckily I was able to pull Craig Larman's "Applying UML and Patterns" book section in google. As quoted
Accessing methods retrieve(accessor method) or set(mutator method)
attributes. In some languages such as Java it is a common idiom to
have an accessor and mutator for each attribute, and to declare all
attributes private(to enforce data encapsulation). The methods are
excluded in the class diagram because of the high noise-to-value ratio
they generate."

Java Getters and Setters are accessor methods. So, yes, they are methods.
Many programmers (Java or otherwise) may feel annoyed at constantly writing getXXX() and setXXX(type t) for all of their private fields, especially if they are basically just one line methods. For this case, there are some annotation libraries (like Lombak) that generate these through the power of metaprogramming and Java annotations and reflection.
However, there are many reasons to explicitly write getters/setters. A good list can be found here. But I really like this answer, too.

Yes, essentially they are methods.
The standard definition for a Java method is as follows;
A method is a set of code which is referred to by name and can be called (invoked) at any point in a program simply by utilizing the method's name.
-https://mathbits.com/MathBits/Java/Methods/Lesson1.htm
So you can consider methods as small programs within a class itself that allows us to fulfill specific tasks, which is also exactly what getters and setters do.

What is the convention with Java beans, and implementing interfaces like Comparable?

Java Beans, as far as I know, should always:
Have only an empty constructor
Have only fields, and getter/setter methods for these fields.
However, I am wondering what the convention is for Java beans to implement interfaces like Comparable? I could leave the java bean pure, meaning absolutely no behaviour, only data, and write a custom comparator-class. Implementing comparable is easier though.
Are there any conventions when it comes to implementing simple common interfaces like Comparable to Java Beans? I cannot find any consequences myself, but it feels like I might breaking some rules, and that there might be something I haven't thought of.

IMHO this question is mostly not about conventions but about the needs.
You are right that separation of bean stuff from any business logic is a good style. I'd add here that this typically a good practice because relationship between your beans and comparators are many-to-many, i.e.
You will probably hold several comparators for one bean class and use them in different contexts
Sometimes you can re-use one comparator for several different classes or make hierarchy of comparators.
However you are right that writing comparison logic into class itself is less verbose and in some cases preferable. The choice completely depends on the author taste and requirements of the application he/she is working on.

Naming the member variable beginning with m_ conforms to javabean style?

Wondering if m_ naming convention conforms to Javabean property style?

well, m_ comes from Microsoft standards, which is based on Hungarian Notation. So that naming pattern isn't recommended by Sun/Oracle.
That's not to say you can't use them in JavaBeans as, javabeans are all about the method signatures, not about the field names, so you can call fields whatever you like.

Well,You can name your Java bean member, using any naming convention. The name is not part of the contract that makes a class, Java Bean.
Your class needs a public contructor,all its properties should be accessible via get and set and the class needs to be serializable.
But then again, I would not suggest using the m_ naming convention. Java Beans convention are used by lots by Java tooling support and more importantly by frameworks like JSF and Web Services (for UI binding and xml marshaling respectively). So apart from confusing some Java developers, who needs to maintain your code, there is also a possibility that some tooling support might not work.

What is meaning of Plain Old Java Object (POJO)?

What does the term Plain Old Java Object (POJO) mean? I couldn't find anything explanatory enough.
POJO's Wikipedia page says that POJO is an ordinary Java Object and not a special object. Now, what makes or what doesn't make and object special in Java?
The above page also says that a POJO should not have to extend prespecified classes, implement prespecified Interfaces or contain prespecified Annotations. Does that also mean that POJOs are not allowed to implement interfaces like Serializable, Comparable or classes like Applets or any other user-written Class/Interfaces?
Also, does the above policy (no extending, no implementing) means that we are not allowed to use any external libraries?
Where exactly are POJOs used?
EDIT: To be more specific, am I allowed to extend/implement classes/interfaces that are part of the Java or any external libraries?

Plain Old Java Object The name is used to emphasize that a given object is an ordinary Java Object, not a special object such as those defined by the EJB 2 framework.
class A {}
class B extends/implements C {}
Note: B is non POJO when C is kind of distributed framework class or ifc.
e.g. javax.servlet.http.HttpServlet, javax.ejb.EntityBean or J2EE extn
and not serializable/comparable. Since serializable/comparable are valid for POJO.
Here A is simple object which is independent.
B is a Special obj since B is extending/implementing C. So B object gets some more meaning from C and B is restrictive to follow the rules from C. and B is tightly coupled with distributed framework. Hence B object is not POJO from its definition.
Code using class A object reference does not have to know anything about the type of it, and It can be used with many frameworks.
So a POJO should not have to 1) extend prespecified classes and 2) Implement prespecified interfaces.
JavaBean is a example of POJO that is serializable, has a no-argument constructor, and allows access to properties using getter and setter methods that follow a simple naming convention.
POJO purely focuses on business logic and has no dependencies on (enterprise) frameworks.
It means it has the code for business logic but how this instance is created, Which service(EJB..) this object belongs to and what are its special characteristics( Stateful/Stateless) it has will be decided by the frameworks by using external xml file.
Example 1: JAXB is the service to represent java object as XML; These java objects are simple and come up with default constructor getters and setters.
Example 2: Hibernate where simple java class will be used to represent a Table. columns will be its instances.
Example 3: REST services. In REST services we will have Service Layer and Dao Layer to perform some operations over DB. So Dao will have vendor specific queries and operations. Service Layer will be responsible to call Which DAO layer to perform DB operations. Create or Update API(methods) of DAO will be take POJOs as arguments, and update that POJOs and insert/update in to DB. These POJOs (Java class) will have only states(instance variables) of each column and its getters and setters.
In practice, some people find annotations elegant, while they see XML as verbose, ugly and hard to maintain, yet others find annotations pollute the POJO model.
Thus, as an alternative to XML, many frameworks (e.g. Spring, EJB and JPA) allow annotations to be used instead or in addition to XML:
Advantages:
Decoupling the application code from the infrastructure frameworks is one of the many benefits of using POJOs. Using POJOs future proofs your application's business logic by decoupling it from volatile, constantly evolving infrastructure frameworks. Upgrading to a new version or switching to a different framework becomes easier and less risky. POJOs also make testing easier, which simplifies and accelerates development. Your business logic will be clearer and simpler because it won't be tangled with the infrastructure code
References : wiki source2

According to Martin Fowler, he and some others came up with it as a way to describe something which was a standard class as opposed to an EJB etc.

Usage of the term implies what it's supposed to tell you. If, for example, a dependency injection framework tells you that you can inject a POJO into any other POJO they want to say that you do not have to do anything special: there is no need to obey any contracts with your object, implement any interfaces or extend special classes. You can just use whatever you've already got.
UPDATE To give another example: while Hibernate can map any POJO (any object you created) to SQL tables, in Core Data (Objective C on the iPhone) your objects have to extend NSManagedObject in order for the system to be able to persist them to a database. In that sense, Core Data cannot work with any POJO (or rather POOCO=PlainOldObjectiveCObject) while Hibernate can. (I might not by 100% correct re Core Data since I just started picking it up. Any hints / corrections are welcome :-) ).

Plain Old Java Object :)
Well, you make it sound like those are all terrible restrictions.
In the usual context where POJO is/are used, it's more like a benefit:
It means that whatever library/API you're working with is perfectly willing to work with Java objects that haven't been doctored or manhandled in any way, i.e. you don't have to do anything special to get them to work.
For example, the XStream XML processor will (I think) happily serialize Java classes that don't implement the Serializable interface. That's a plus! Many products that work with data objects used to force you to implement SomeProprietaryDataObject or even extend an AbstractProprietaryDataObject class. Many libraries will expect bean behavior, i.e. getters and setters.
Usually, whatever works with POJOs will also work with not-so-PO-JO's. So XStream will of course also serialize Serializable classes.

POJO is a Plain Old Java Object - as compared to something needing Enterprise Edition's (J2EE) stuff (beans etc...).
POJO is not really a hard-and-fast definition, and more of a hand-wavy way of describing "normal" non-enterprise Java Objects. Whether using an external library or framework makes an object POJO or not is kind of in the eye of the beholder, largely depending on WHAT library/framework, although I'd venture to guess that a framework would make something less of a POJO

The whole point of a POJO is simplicity and you appear to be assuming its something more complicated than it appears.
If a library supports a POJO, it implies an object of any class is acceptible. It doesn't mean the POJO cannot have annotations/interface or that they won't be used if they are there, but it is not a requirement.
IMHO The wiki-page is fairly clear. It doesn't say a POJO cannot have annotations/interfaces.

What does the term Plain Old Java Object (POJO) mean?
POJO was coined by Martin Fowler, Rebecca Parsons and Josh Mackenzie when they were preparing for a talk at a conference in September 2000. Martin Fowler in Patterns of Enterprise Application Architecture explains how to implement a Domain Model pattern in Java. After enumerating some of disadvantages of using EJB Entity Beans:
There's always a lot of heat generated when people talk about
developing a Domain Model in J2EE. Many of the teaching materials and
introductory J2EE books suggest that you use entity beans to develop a
domain model, but there are some serious problems with this approach,
at least with the current (2.0) specification.
Entity beans are most useful when you use Container Managed
Persistence (CMP)...
Entity beans can't be re-entrant. That is, if you call out from one
entity bean into another object, that other object (or any object it
calls) can't call back into the first entity bean...
...If you have remote objects with fine-grained interfaces you get
terrible performance...
To run with entity beans you need a container and a database
connected. This will increase build times and also increase the time
to do test runs since the tests have to execute against a database.
Entity beans are also tricky to debug.
As an alternative, he proposed to use Regular Java Objects for Domain Model implementation:
The alternative is to use normal Java objects, although this often
causes a surprised reaction—it's amazing how many people think that
you can't run regular Java objects in an EJB container. I've come to
the conclusion that people forget about regular Java objects because
they haven't got a fancy name. That's why, while preparing for a talk
in 2000, Rebecca Parsons, Josh Mackenzie, and I gave them one: POJOs
(plain old Java objects). A POJO domain model is easy to put together,
is quick to build, can run and test outside an EJB container, and is
independent of EJB (maybe that's why EJB vendors don't encourage you
to use them).

There is an abundance of posts that are half correct and half incorrect. The best example of the correct interpretation is given by Rex M in their answer here.
[POJO are classes] that doesn't require any significant "guts" to make
it work. The idea is in contrast with very dependent objects that have
a hard time being (or can't be) instantiated and manipulated on their
own - they require other services, drivers, provider instances, etc.
to also be present.
Unfortunately, these very same answers often come along with the misunderstanding that they are somehow simple or often have a simple structure. This is not necessarily true and the confusion seems to stem from the fact that in the Java (POJO) and C# world (POCO) business logic is relatively easily modeled especially in the web application world.
POJO's can have multiple levels of inheritance, generic types, abstractions, etc. It just so happens that this isn't required in the majority of web applications as business logic doesn't necessitate it - alot of the effort goes into databases, queries, data transfer objects and repositories.
As soon as you step out of line with simple web apps, your POJO's start looking a lot more complex. E.g. Make a web app that assigns taxi's to user schedules. To do this, you need a graph coloring algorithm. To color the graphs, you need a graph object. Each node in the graph is a schedule object. Now what if we want to make it generic so that coloring the graph can be done not only with schedules but other things as well. We can make it generic, abstract and add levels of inheritance - almost to the point of making it a mini library.
At this point though, no matter its complexity, its still a POJO because it doesn't rely on the guts of other frameworks.

A Plain Old Java Object (POJO) that contains all of the business logic for your extension.
Exp. Pojo which contains a single method
public class Extension {
public static void logInfo(String message) {
System.out.println(message);
}
}

Are these synonymous, a subset of each other or completely different?

Are the notions mentionned in the question title synonymous to a certain degree? Where do the main differences lie (context, structure, ...) and can one be considered a subset of another? Here's some brief definitions taken from Wikipedia.
POJO (Plain Old Java Object)
Wikipedia
In computing software, POJO is an
acronym for Plain Old Java Object. The
name is used to emphasize that a given
object is an ordinary Java Object, not
a special object, and in particular
not an Enterprise JavaBean. The term
was coined by Martin Fowler, Rebecca
Parsons and Josh MacKenzie in
September 2000:
"We wondered why people were so against using regular objects in their
systems and concluded that it was
because simple objects lacked a fancy
name. So we gave them one, and it's
caught on very nicely."
Java Bean Wikipedia
JavaBeans are reusable software
components for Java that can be
manipulated visually in a builder
tool. Practically, they are classes
written in the Java programming
language conforming to a particular
convention. They are used to
encapsulate many objects into a single
object (the bean), so that they can be
passed around as a single bean object
instead of as multiple individual
objects. A JavaBean is a Java Object
that is serializable, has a nullary
constructor, and allows access to
properties using getter and setter
methods.
Value Object Wikipedia
Data transfer object (DTO), formerly
known as value objects or VO, is a
design pattern used to transfer data
between software application
subsystems. DTOs are often used in
conjunction with data access objects
to retrieve data from a database.
Business Object Wikipedia
A business object is a type of an
intelligible entity being an actor
inside the business layer in a
n-layered object-oriented computer
program.
Related:
Difference between DTO, VO, POJO, JavaBeans?
What is the difference between a JavaBean and a POJO?
DDD: what's the use of the difference between entities and value objects?

Not all of these classifications are related. Here's my understanding:
POJO is what its name suggests - a plain old Java object. There's nothing special about it. And this is exactly what we want to convey when we say that an object is a POJO. Today most applications are using some kinds of underlying frameworks, and with the frameworks come requirements on the objects that will integrate with the framework - the object must implement an interface or extend a class. When we say an object is a POJO, we mean to say it is just an ordinary object and has no dependencies on any framework.
A JavaBean is a java class that follows certain conventions as described in your question. Such objects are often mandated by certain frameworks which use reflection to find out the properties (accessible through getters/setters) of the object and manipulate them e.g. beans exposed to JSPs, Spring beans etc. The good thing about JavaBeans is that they are still POJOs. Although they follow certain conventions, the conventions are not defined by any particular framework but are rather defined by Sun Javabean standard and the classes are still plain Java classes with no ties to any third party framework's classes or interfaces.
Business Objects refer to objects that represent your business domain entities. These usually reside in your business layer - the layer where all the business logic is. These objects usually map to persistence store entities e.g. tables. These objects could be POJOs, JavaBeans, EJBs etc.
Value objects are a type of design pattern. In some small web applications, you have the option of using your business objects in the web layer as well. However, in larger applications or J2EE applications, you define value objects to move information from the business layer to the web layer. That's why they are also called Data Transfer Objects (DTOs). These objects usually have only the attributes that are needed in the web layer and leave the attributes of business objects that were meant for business layer consumption behind. They may also have "computed" attributes that are generated in the business layer. Using this patterns helps decouple the business and web layers.

Here's my take:
Business objects is a generic term
for the abstract idea that
represents your problem. You can
implement them in any language. In
Java, you have additional choices to
make, because they can be POJOs or
EJBs, mutable or immutable.
Value objects or DTOs are used to ferry data between layers. They're usually immutable. They can be implemented as POJOs or Java Beans. Think of them as another subset of POJOs.
A Java Bean conforms to the original Sun specification. They were intended to provide an interface that would allow them to be plugged into a VB-style IDE with ease. Think of these as a subset of POJO.
People sometimes get confused about the difference between Java Beans and Enterprise Java Beans. Java Beans are part of the original Java 1.0 spec, intended to be like VB components (remember "Bean Box"?). Enterprise Java Beans were a spec that followed that described how special Java objects would implement specific interfaces to interoperate with a Java EE app server. The app server was a transaction monitor for a distributed component architecture that would handle threading, persistence, pooling, object lifecycle, messaging, naming, etc. EJBs are a very special subset of Java objects that work only within the context of a Java EE app server.
A POJO can be implemented to conform to the Java Bean standard, but it's not a requirement. Any Java object qualifies as a POJO. It was originally meant to distinguish them from EJB version 2.0, which required several interfaces in order to interoperate with the Java EE app server properly.

The questions is whether it's a mistake to use some of these as synonyms (like I've heard some people do) and if a given classification can be considered as a subset or another.
It is a mistake to use these terms as synonyms. They clearly have distinct meanings. The quoted definitions (and those provided in other answers) make this clear.
However, if it is often valid to use many (or even all) of these terms to describe the same object or objects. It is all a matter of perspective; i.e. what aspect of the object(s) you are trying to emphasize.

Synthesis (from answers given):
POJO: An ordinary object with no dependencies towards any framework. It can be adapted to conform to the Java Bean standard without being a requirement as such.
JavaBean: Object conforming to the Sun JavaBean or Java 1.0 specification (refer to "Bean box"). They were originally intended to provide an interface so they could be plugged into a VB-style IDE with little difficulty. Can be considered as a subset of POJOs and remain independant of frameworks. It can employ certain mecanisms such as reflection to access properties.
Enterprise Java Bean: These shouldn't be confused with Java Beans. With the simplifications brought about with version 3.0, EJBs can be considered as equivalent to a POJO. EJB in itself is a specification describing special Java Objects that can interoperate with a Java EE server. The server as such acted as a transaction monitor in the context of a distributed component architecture handling things such as threading, persistence, pooling, object lifecycle, messaging and naming. As such an EJB can be viewed as a very special subset that used in the contect of a Java EE application server.
Business object: Theoretical concept or abstract idea that helps to represent a given problem. It represents business domain entities and resides in the business layer of an application. They can be mapped to entities in the context of persistance. The object can be a POJO/JavaBean/EJB and be either mutable or immutable.
Value object/Data Transfer Object: Employs a design pattern which helps to decouple the business and web layers. This is to suit the context of large applications where objects can transit between layers (the business and web layer for example). They're usually immutable in nature and can either be formated as POJOs or Java Beans. One specificity is that they can contain computed attributes that are generated in the business layer.
P.S: Marked as community wiki so feel free to edit.