Ajax is not a technology in itself, but a term that refers to the use of a group of technologies together.

Traditional web applications essentially submit forms, completed by a user, to a web server. The web server responds by sending a new web page back. Because the server must submit a new page each time, applications run more slowly and awkwardly than their native counterparts.

Ajax applications, on the other hand, can send requests to the web server to retrieve only the data that is needed, usually using SOAP or some other XML-based web services dialect.

On the client, JavaScript processes the web server response.

The result is a more responsive interface, since the amount of data interchanged between the web browser and web server is vastly reduced. Web server processing time is also saved, since much of it is done on the client.
(Source: Wikipedia)

Here is Eddie Traversa excellent introduction to Ajax, which not only provides an excellent introduction for the layman, but further dives into analyzing specific Ajax coding techniques and examples.