The main purpose you use AJAX for is to get data from a web server and display it within your web page without having to reload the page. From the user's perspective, this improves the experience because nothing frustrates web users more than waiting a few seconds for a page to load.
If every link and button you press on the web page causes the page to reload, that totals to a lot of time the user is looking at your browser waiting for information to be displayed.
Of course, its not just about fetching data. You can also send data back to the server for processing. Take the example on this web page. When you click on the "Reply to this Discussion" button, the page does not perform a full reload. That button, once clicked, takes the content you typed in the post textbox and sends it back to the web server and the information is stored in the database and updates the area of the page that you were typing in. That's done via AJAX.