The best use of AJAX is where it is used to send small payloads. Here is a simple example.
I load a page that contains information about stock. It has graphs, charts, company information and it also displays the share-price. Every 30 seconds, I make an AJAX request that gets the updated share-price and changes it on the page.
Without AJAX, I might decide to refresh the entire page every 30 seconds, but with AJAX, I can just make a lightweight request to get the tiny bit of information I need.
Using AJAX to submit forms isn't always the best bet. Asides from not really giving you a clear advantage over posting the form normally, you break conventions such as browser history (although some browsers now include JavaScript "states" as pages in the history).
When using AJAX, you need to handle the task of telling the user if something has gone wrong. You can do this with jQuery by specifying what should happen on error, but many people forget to do this and the end user is blissfully unaware of any problem.
Other issues to watch out for are any JavaScript errors that may prevent your events from firing - or if JavaScript is disabled, in either case ensuring that the form can submit normally before you add the AJAX code is the safest option.
