Stateless vs Stateful

A stateful app is one that stores information about what has happened or changed since it started running. Any public info about what "mode" it is in, or how many records is has processed, or whatever, makes it stateful.

Stateless apps don't expose any of that information. They give the same response to the same request, function or method call, every time. HTTP is stateless in its raw form - if you do a GET to a particular URL, you get (theoretically) the same response every time. The exception of course is when we start adding statefulness on top, e.g. with ASP.NET web apps :) But if you think of a static website with only HTML files and images, you'll know what I mean.

Stateless means there is no memory of the past. Every transaction is performed as if it were being done for the very first time.

Stateful means that there is memory of the past. Previous transactions are remembered and may affect the current transaction.

Stateless:

// The state is derived by what is passed into the function

function int addOne(int number)
{
    return number + 1;
}

Stateful:

// The state is maintained by the function

private int _number = 0; //initially zero

function int addOne()
{
   _number++;
   return _number;
}

Refer from: https://softwareengineering.stackexchange.com/questions/101337/whats-the-difference-between-stateful-and-stateless

I suggest that you start from a question in StackOverflow that discusses the advantages of stateless programming. This is more in the context of functional programming, but what you will read also applies in other programming paradigms.

Stateless programming is related to the mathematical notion of a function, which when called with the same arguments, always return the same results. This is a key concept of the functional programming paradigm and I expect that you will be able to find many relevant articles in that area.

Another area that you could research in order to gain more understanding is RESTful web services. These are by design "stateless", in contrast to other web technologies that try to somehow keep state. (In fact what you say that ASP.NET is stateless isn't correct - ASP.NET tries hard to keep state using ViewState and are definitely to be characterized as stateful. ASP.NET MVC on the other hand is a stateless technology). There are many places that discuss "statelessness" of RESTful web services (like this blog spot), but you could again start from an SO question.

The adjective Stateful or Stateless refers only to the state of the conversation, it is not in connection with the concept of function which provides the same output for the same input. If so any dynamic web application (with a database behind it) would be a stateful service, which is obviously false. With this in mind if I entrust the task to keep conversational state in the underlying technology (such as a coockie or http session) I'm implementing a stateful service, but if all the necessary information (the context) are passed as parameters I'm implementing a stateless service. It should be noted that even if the passed parameter is an "identifier" of the conversational state (e.g. a ticket or a sessionId) we are still operating under a stateless service, because the conversation is stateless (the ticket is continually passed between client and server), and are the two endpoints to be, so to speak, "stateful".