Ajax is a collection of technologies that allow uses to interact with a web site, and the web site to interact with a server, without post-backs and server round trips. Thus providing a richer user experience. Before I go on, try it out and see what you think.
Google maps : http://maps.google.co.uk - great dynamic maps you can drag about, with neat integration to the search engine. As you drag to other parts of the map, Ajax gets the new parts of the map for you from the server.
A9 search : http://a9.com - Amazons search engine, try the column buttons on the left hand side to see Ajax in action
Google Suggest: http://www.google.com/webhp?complete=1&hl=en - as you type in the search box, Ajax technology pops up matching search phrases.
Poetry fun: http://www.brock.ac.uk/blogs/jon/poetry/ - magnetic poetry - get two or more people on the site to see the interaction.
Jesse James Garrett has a great article on Ajax here: http://www.adaptivepath.com/publications/essays/archives/000385.php. I quote:
The biggest challenges in creating Ajax applications are not technical. The core Ajax technologies are mature, stable, and well understood. Instead, the challenges are for the designers of these applications: to forget what we think we know about the limitations of the Web, and begin to imagine a wider, richer range of possibilities.
This is exciting technology that works. It has the potential to bring the richness of a Windows UI to the web. Google are obviously investing heavily in it. It's one to watch...
******** UPDATE
Asp.net 2.0 supports this technology via Client Callbacks - see http://www.dotnetjunkies.com/Tutorial/E80EC96F-1C32-4855-85AE-9E30EECF13D7.dcik for an example