AJAX is an acronym that stands for "Asyncronous Javascript and XML". For those less technically inclined, AJAX is a technique, not a product or a brand.
One of the classic drawbacks to building a web application interface is that once a page has been downloaded to the client, the connection to the server is severed. Any attempt at a dynamic interface involves a full roundtrip of the whole page back to the server for a rebuild. This process tends to make web apps feel inelegant and unresponsive. Ajax offers a solution using JavaScript and the XMLHttpRequest object. AJAX allows a website to communicate with the server without having to load a new page into the browser. By communicating with the server while the page is still loaded, data can be retrieved and dynamically inserted and removed from the existing page, or altered as required. A popular Ajax app is used with Google maps.
However, the key challenges to accessiblity include: AJAX apps require browsers enabled with JavaScript and will not perform in web browsers and devices that don't support JavaScript. The Web Content Accessibility Guidelines also require that web applications function when JavaScript is disabled or not supported. Additionally, AJAX requires that the XMLHttpRequest be supported.
The current solution to these problems is to either provide a non-AJAX alternative or to allow the AJAX application to continue to function if JavaScript and XMLHttpRequest are not supported. So, when users allow Ajax to be implemented, the web page will communicate with the web server without requiring reloads for more responsive user interfaces.