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What is the purpose of web server in a local web application?

I am sure this sounds silly, but for a beginner like me, this is cooking my brain up and I am not able to proceed in my quest without clearing this.
Lets say I am building a simple java command line calculator application that takes 2 numbers and an operator as input and returns the operation's result back to the user. Now, I want to build GUI on top of it (similar to like an online calculator). For this, I decided to build a web application so that I can open the application on my browser and use it seamlessly. It leads to the following questions in my head
Will I have to build a web server in java to allow communications between the front end/GUI and the back end? There is no communication with any other networks, its strictly local; yet I am having to create a web server to make this work and that is confusing me. Or do I need one because I decided to build a web application meant to be opened in a web browser? Or is the reason something else entirely?
If I instead decided to build a windows/android/mac/ios application, would I still to bake a web server for communicating? Or can I use something like swings (I know its really primitive) to do this which essentially would allow the GUI to directly communicate with the business logic?
I know front-end languages like javascript allow me to code the entirety of the calculator in itself, and as a result eliminate the need of a web server. However, what if I was building relatively complex application with a backend database, like spotify? (within the same constraint that the application doesn't need to communicate with other applications and all data is stored locally) Would this solution still work?
In brief, I am failing to understand the purpose of building an entire client-server infrastructure even though the entire thing is local and offline(if that makes sense?). Or is my very basic understanding of client-server flawed?

If you want to use a web browser as the tool to interface with your Java app, then you need a web server. The browser is simply an app that tries to make a network connection with another app, and passes some text as defined by the HTTP protocol.
You would have a choice of three scenarios for this:
Write an app that accepts network connections, processes HTTP, and sends back a response.
Write a Java app that uses the very basic web server built into Java 18 and later. Note that this web server is not intended to be a feature-rich or commercial-grade server.
Write a Jakarta Servlet class that runs on top of a Servlet container with a web server. For your needs, either Apache Tomcat or Eclipse Jetty would work well as both the Servlet container and the web server.
For your scenario, the middle option using Java JEP 408 seems most appropriate and easiest.
You said:
There is no communication with any other networks, its strictly local; yet I am having to create a web server to make this work and that is confusing me.
A Border Collie dog herds sheep. That is its most basic basic core mission. Those sheep can be herded across the hills of a mountain range or they can be herded locally within your own farm.
A web browser makes network connections. That is its most basic core mission. Those network connections can be made over the actual network or they can be made locally within a single computer ( a “localhost”).
You said:
Or do I need one because I decided to build a web application meant to be opened in a web browser?
Yes a web application accessed by a web browser needs a web server, by definition.
You said:
If I instead decided to build a windows/android/mac/ios application, would I still to bake a web server for communicating?
Or can I use something like swings (I know its really primitive)
There is nothing “primitive” about Swing. It is a fully-developed feature-rich GUI framework. Swing comes built in with every JDK. And Swing will be supported for many years to come.
However, the design and features of Swing may or may not suit your tastes. And Swing is now in maintenance-mode.
An alternative is JavaFX, now actively developed as the OpenJFX libraries.
To use JavaFX, you must either add the OpenJFX libraries to your project or else deploy to a JDK that comes bundled with the OpenJFX libraries (ZuluFX, LibericaFX, etc.).
to do this which essentially would allow the GUI to directly communicate with the business logic?
Yes the GUI and and your calculator business logic would all be plain Java classes, all peers, all running together within the same JVM.
You said:
I know front-end languages like javascript allow me to code the entirety of the calculator in itself, and as a result eliminate the need of a web server.
Yes. You could cook up JavaScript code to implement your little calculator. This JavaScript could be placed within the text of a file with the HTML of a web page. That web page file could be opened locally by the web browser. Your HTML and JavaScript would render. No web server is needed in this scenario.
But this scenario eliminates Java, and you said you want to (a) write your calculator in Java, and () use a web browser. So we go back to the three options listed above.
You said:
However, what if I was building relatively complex application with a backend database, like spotify? (within the same constraint that the application doesn't need to communicate with other applications and all data is stored locally) Would this solution still work?
Yes you could deploy a database server on your local machine. Then you could find and use a JavaScript binding tool to talk to that database.
But then you would not be using Java and all its goodness. For that scenario I would choose to write a JavaFX app with an embedded Java-based database engine such as H2. This would result in a single all-in-one double-clickable-app solution.
But that would be my choice based on my own preferences and skill set. Other people might choose other solutions.
the purpose of building an entire client-server infrastructure even though the entire thing is local and offline
👉 The “offline” part may be distracting you.
Conventional computers with modern operating systems such as macOS, BSD, Linux, Windows, etc. are always “online” in that they all maintain a networking stack. That network stack can be used locally by processes within the one computer to communicate with each other. Whether you happen to have an Ethernet cable plugged in, or WiFi turned on, makes little difference to the computer. The networking stack in the OS is still available and active regardless of outside network access.
So client-server architecture works just as well within a single computer as it does between computers. (Actually, it’s works faster locally, as hopping on and off the network is terribly slow.)
Why choose client-server architecture for a local app? Either:
You are skilled in, and prefer, the tools for a client-server architecture.
You want to eventually move from local-only to being networked.
If neither of those two is true, then the JavaFX with H2 solution I mentioned is likely a better fit for a Java programmer.
By the way, let me put in a plug for Vaadin Flow, a GUI framework for building web apps in pure Java.

It's not completely necessary to bake web server to host your application even with embedded tomcat application you can host your application. However, main purpose of webserver usage is for setting up your reverse proxy so that your application behind that won't be exposed and your webserver will be acting as endpoint to outside requests.
In case for hosting application in local webserver is not at all required webserver will come handy when your application hosted in production environment.

How do I make a simple web app with log-in?

I have been doing android programming for a while now but only as a hobby. I know the basics of java, and can say I have a solid understanding of PHP and MySQL (I once followed a tutorial that showed how to create a very basic content management system). I've been wanting to expand my knowledge beyond the simple android apps I've made and recently had an idea for an android app. In this app, the user would create a Username and password the first time it is ran. From then, the user can fill out a form. From what i know so far, the valuesof this form can be stored in a MySQL database. So basically every user needs to have their own set of variables stored (which are not a lot). As I been looking around, i think there are many ways to create a web app, and there are different frameworks for doing so. I read I can create a web app with log-in, using ASP.NET. Can this be done using java? I just need some general guidance. I want to make the web app standalone, and then focus on creating an app for android that uses it.

I think good platform for your kind of case is to use Google App Engine (GAE). It provides platform to do your web-service with Java (or python if you prefer). It is also free for low amounts of traffic (like your service) and they have really good tools to manage the site (check the database entries, usage statistics, etc.).
Google has written a good set of tutorials to build webservice with Java in GAE:
http://googcloudlabs.appspot.com/
AppEngine documentation main page:
http://code.google.com/intl/fi-FI/appengine/
Signup here:
https://appengine.google.com/

I think you may want to look at JavaServer Pages.

Multiple Frontends - Java Backend

I just started working on a project which will about making a big website.
With big i mean:
Webshop
Forum
Normal Website (Information pages etc..)
At first i just wanted to pick a decent open-source webshop and just built my site around it.
But then i started thinking about how to expand etc.
I started to think how i would like this website to be acces from a mobile phone. Not just with a browser but with an App. (I have decent experience in making apps for Android & Iphone). So the real question is:
Would it be smart to make the "Core" of my whole website in Java and use services to acces it and thus allowing different frontends to use the same "Core". Like:
- PHP for browser frontend
- Java (android) for the android App
- Objective C for the IPhone App.
And let them all just communication to the "Core" through REST (Json).
What will be the advantages / disadvantages with this approach and will there be a significant delay in rendering eg. a webpage (http request to php, then php making calls to java server (different physical server) then accessing the database and then returning it all, so php can format it to HTML).
Hope hearing some answers or suggestions!

I would skip the Java layer and make the "core" PHP. There is a wealth of tools and frameworks (like Zend FW) for building thoses components in PHP. Design your application around a REST interface and allow your mobile apps to use REST.
Better yet, use Rails. It's so easy to design and set up a REST interface.

I think your solution is “smart”. The only changes I suggest are:
Use the same platform to build both the "core" and the "browser frontend." This way you will have more productivity.
Keep the "core" and "browser" frontend on the same server, if possible in the same process. Unless you have a reason do it differently. This will reduce problems with performance, latency and so on.

If you're already familiar with Java I would suggest you look at something like GWT (or GWT + third-party libraries like Ext GWT or Smart GWT) for your front-end. I don't see the point in picking another language for the front-end unless you really want to learn something else (e.g. PHP as you suggest in your question).
I think the rest of your approach is sound (i.e. Java on the backend, providing RESTful services, etc.)

Can i create Web App to be used on smart-phones without internet ? How?

I want to make cross - platform app which could be used for different smart-phones, and been advised to make 'Web App'.
Can 'web app' work without internet?
How to do it, which programing language to use?
thanks

Oh sure. Bundle all your HTML, JS anc CSS with your app, and have a thin Java/Android app over these assets. If you are not making any calls to fetch or show external HTML/pages, you don't need internet.

Most likely no.
While it is possible to run "web apps" without access to the internet (you can simply execute Javascript locally from some browsers), it would be
(Subjectively) not particularly useful.
Not supported by many devices, whom (like the iPhone) do not allow the storage and execution of arbitrary, non-native code on the device.
However, some frameworks do exist that hold the idea of "web apps" can be a distribution model. For example, PhoneGap allows you to code in HTML5/Javascript, and distribute your app to multiple platforms.

Yes, it can work offline once downloaded from web. Use just any regular programming language for web application development. If you are good at programming, I recommend ruby-on-rails or ASP.NET MVC with Jquery mobile.

You might want to look into frameworks like:
PhoneGap: http://www.phonegap.com/
jQuery Mobile: http://jquerymobile.com/
GWT Mobile: http://code.google.com/p/gwt-mobile-webkit/
Your question is very broad so it's hard to give a useful answer. Essentially when making a web app you'll normally be working with javascript, css, and html... though there are frameworks which will let you write your app in Java/Python/Ruby and then have those languages create your app for you. Keep in mind this is mostly all front facing stuff though. The actual web app will still need to have a backend written in a programming language that you understand (hopefully).
Good luck :).

Check out the jQuery Mobile API - I started using it recently and it has proved to be a viable option. Check this out.

https://developer.mozilla.org/en/Using_Application_Cache with an application cache your Web App can be used offline. It will cache the resources you specify to disk.
So you can write a Javascript + HTML webpage/app and use it both on and off line.
You can cache/store:
HTML
CSS
Javascript
Images
GET Requests

What applications or libraries are available for screen scraping applications on Windows?

We develop an application that clients use side-by-side with in-house custom software. One of our clients has requested that we investigate the use of screen scraping software to provide a sort of integration layer between the two applications.
Our software is written as a Java (QT Jambi) desktop application, which hosts a web browser control, that runs the application in Javascript.
The client software is a Windows desktop thick client. Unsure what it is written in (managed .NET or unmanaged win32? Maybe even Java).
We need to get data out of our application, and input it into their application. Their application can not be modified under any circumstances. They have suggested using screen scraping to read the information from our app, and input into text fields within their app.
Are there any screen scraping applications that can be used for this purpose? Are there any libraries that use some kind of OCR tech written in .NET? I think I want to avoid the type that hook into WM_Messages, but I'm unsure why.
Any pointers in the right direction would be appreciated.

What about making Windows API calls to interact with the GUI controls directly? If its written in .NET, this should be rather straightforward.