AJAX is a web development technique used for creating interactive web applications. The idea is to make web pages feel more responsive by exchanging small amounts of data with the server behind the scenes. This means that the entire web page does not have to be reloaded each time the user requests a change resulting in increases in the web page's interactivity, speed, functionality, and usability.
You can see it in action by going to Sleeptime and clicking on the MP3 blue button on the right hand side (don't worry you don't have to buy anything). One way of looking at it is that it enables sites to operate from top to bottom in a funnel rather than the traditional left to right process.
Or go visit UPS and have a look.
If you really want to know, AJAX is shorthand for Asynchronous JavaScript and XML. Ajax is asynchronous in that loading does not interfere with normal page loading. JavaScript is the programming language in which Ajax function calls are made. Data retrieved using the technique is commonly formatted using XML. There you go!