AJAX is used in cases where you can speed-up user actions by avoiding useless page refresh. In general:
Use AJAX when an action of the user affects a small part of the page and the action is done regularly.
Don't use it when an action of the user changes nearly the whole page or the action is done rarely.
Since you're complaining in your comment of the answer by DougM to be unable to find an example, here's an example which considers different scenarios of an application, indicating for each whether AJAX can be used and what criteria should be taken in account to make the choice.
Let's take a web page where users can view and add post-its (which would be useful in a context when they need to take short notes, like a phone number, in order to access them later from everywhere). For the sake of simplicity, we will not discuss registration/logon.
First, the user arrives. This is an ordinary request, since there is no page which can be used as a base for AJAX. So no AJAX here.
Then, the user fills an empty post-it and submits the new content. Here, AJAX can be useful: instead of forcing the user into the page reload, sending again all the menus and logo at the top and copyright notice and links at the bottom, etc. A more elegant way would be simply to shake the post-it to indicate that it's posted, or change its appearance in some way, and voila.
If the post-it is removed, AJAX can be useful again. Instead of refreshing the whole page, just remove the concerned post-it from the list. The situation is similar when the user removes the post-it, then clicks on "Undo removal" link which appears for a few seconds.
Let's assume now that not every post-it is displayed on the page, i.e. there is a "Page 1 · 2 · 3 · … · 16" bar. When the user switches from page to page, you might use AJAX, or you may avoid using it. In fact, loading all the post-its on a page may take some time: nearly the same time as loading the whole page with top logo and menus and bottom copyright notice.
Finally, what about a user who removed a post-it few days ago, and now wants to recover it? No matter how much content is changed, you don't use AJAX here. In fact, the feature will be accessed by only a few users, and not very often. Implementing AJAX and adding complexity for a feature which is barely used isn't worth it.