It seems some web architects aim to have a stateless web application. Does that mean basically not storing user sessions? Or is there more to it?
If it is just the use
From a developer's perspective, statelessness can help make an application more maintainable and easier to work with. If I know a website I'm working on is stateless, I need not worry about things being correctly initialized in the session before loading a particular page.
From a user's perspective, statelessness allows resources to be linkable. If a page is stateless, then when I link a friend to that page, I know that they'll see what I'm seeing.
From the scaling and performance perspective, see tsters answer.