For people who don't know about AJAX, Wikipedia gives good info. Ajax in one line is a technology by which you can send and receive data without the page being reloaded, so it looks more 'dynamic'. I guess this could be the dummy way of explaining it.
I wanted to know more about AJAX and learn how to code with AJAX. Just the basic stuff. This is what I found:
Getting Started with AJAX - Author Aaron Gustafson takes you through writing a basic AJAX application (An address book). Lots of code there.
Mastering AJAX - This is a pretty good tutorial which gives you the nitty gritty basics, talks about cross browser compatibility which is really important with XmlHTTPRequest object in IE (stupid MSFT :-p) What I like about this tutorial is that it explains info in a very easy to understand manner and it also says why you need it. (Part 1 (introduction) of the tutorial is here)
Using AJAX with PHP and mySQL - Being a PHP Developer myself, this tutorial is really well written and very complete. Step by step instructions on how to go about working with the XMLHttpRequest object too.
Another good tutorial by John Wiseman on Creating a mySQL connection with PHP and Ajax.(If the code seems too small copy and paste it intoyour favorite editor, or hit Ctrl + '+' on Firefox to increase font size)
Lastly, Max Keisler has a very comprehensive list Ajax tutorials, resources - from beginner to advanced.