AJAX (Asynchronous Javascript And XML) is a cluster of techniques for building browser-side interfaces to websites. A number of very well known websites such as Google Maps use AJAX to achieve a responsive GUI.
Ajax and like techniques have great potential to make web site user interfaces more responsive, but there is potential here to conflict with architectural principles of the web http://www.w3.org/TR/webarch/. In particular, the relationship between a URI used to access a web page and the resource whose state is displayed should not be violated. Note that when a user interacts with a page using Ajax (or similar techniques), the URI of the displayed page does not change, so the underlying resource should similarly be maintained. The exact nature of the relationship between a URI and the corresponding resource is subject to considerable debate, but a useful touchstone might be: if a page using Ajax is bookmarked at any stage during a user interaction, can the URI thus obtained be used to retrieve something clearly related to what is displayed when the bookmark was saved? -- GrahamKlyne
Does modifying the URL in the address bar as the user navigates count? Is this possible? What about providing links in the page to bookmark?
The XMLHttpRequest object, which I understand to be at the core of AJAX in internet explorer, is an ActiveX object, and some strict security configurations prevent it from working. There has been some work to emulate it in JavaScript and IE DOM, and IE7 allegedly will no longer implement the object in this way; presumably part of "drastically reducing the internal attack surface" -- RussellJones
See also:
http://www.adaptivepath.com/publications/essays/archives/000385.php (Google snippets this as "Seminal article which popularised "AJAX" as a term")
http://www.javaworld.com/isearch?qt=AJAX&x=0&y=0&site=javaworld&ms=1&tid=1&st=1&rf=0 javaworld articles on AJAX
http://www.onlamp.com/pub/a/onlamp/2005/06/09/rails_ajax.html - introductory article on rapid developent of Ajax applications with Ruby and Rails
http://www.turbogears.org/ - claims to support Ajax for Python web applications using Mochikit. (An implication here is that a good Javascript HTTP+XML library is part of the key to creating Ajax applications.)
MochikitNotes Mochikit notes - I'm doing some real-time control applications in Javascript (!), and here are a few things I've learned that are not clear in the main Mochikit documentation.
http://www.xml.com/pub/a/2005/02/09/xml-http-request.html an article using PHP on the server side
http://goog-ajaxslt.sourceforge.net/ - the google sponsored implementation of XSLT and X-PATH in javascript
http://rajshekhar.net/blog/archives/85-Rasmus-30-second-AJAX-Tutorial.html - Rasmus' 30 second AJAX Tutorial - a "learn by doing" example
http://ajaxpatterns.org/Main_Page - has some articles covering more advanced Ajax techniques.
Some other links (not yet checked):

