Ajax is best understood by relating it to the way a normal web page is accessed by a user. With a web browser, a user issues a "request" to see a specific web page. That request is sent to the web server. That server gathers those requested files and sends them to the user's computer as a "response" to the "request," and the browser then converts them to a visual form and displays them for the user. This is where Ajax can come in. Instead of requesting that the entire page be recreated, the creator of the web site can make it so that just the new content is requested from the web server.
The way this works is that user action triggers an asynchronous request to the web server for JUST a specific section of the web page. It is asynchronous in that it is not tied to the full response for a web page.