Ajax is not a programming language, nor does it stand for "Asynchronous JavaScript and XML" as some people say. The first person to actually refer to the use of JavaScript in this way claims it should just be called Ajax. I have always like to believe it does stand for that though as it then validates having another acronym (AJAH) for requests to HTML pages.
The first time I encountered Ajax was at a Carson Workshop session with Thomas Fuchs in London and it was amazing to see just what was possible with existing technology. The idea of Ajax is that it makes a connection to a page, service or other which is capable of accepting GET or POST requests and then uses this to make live server-side interactions without refreshing the page. This method also allows updates to take place by using content from the response text.
The advent of actually naming this method has caused a whole new wave of design patterns across the net which have in some cases enriched the users experience, yet in others have just used it for the sake of it. One common problem some sites have is that they don't degrade gracefully - if JavaScript is disabled they cease to function, and many developers and accessibility guru's consider this to be bad. Whilst on the subject of accessibility major site owners should be aware of accessibility laws in their country and should make sure that their site is still functional to people who use screen readers. In the UK for example the law states they must be accessible, and sites hosted here can end up with a hefty fine when they are not. In actuality I don't believe there have yet been any examples of court cases based on this, but it is something keen Ajax advocates should be aware of.
Personally I'm quite fond of Ajax and have used it extensively on many projects and have posted articles on the topic. It's always interesting to see what new ways people can come up with for using it.