XMLHttpRequest() is an object (a constructor function) that allows you to send HTTP requests from JavaScript. Historically, XMLHttpRequest (or XHR for short) was introduced in IE and was initially implemented as an ActiveX object. Starting with IE7 it is a a native browser object, the same way as it is in Firefox, Safari, and Opera. The common implementation of this object across browsers gave birth to the so-called AJAX applications, where it is no longer necessary to refresh the whole page every time you need new content. With JavaScript, you can make an HTTP request to the server, get the response and update only a part of the page. In this way you can build much more responsive, desktop-like web pages.
AJAX stands for Asynchronous JavaScript and XML.
Asynchronous because after sending an HTTP request your code doesn't need to wait for the response, but it can do other stuff and be notified (through an event) when the response arrives.
JavaScript—well, it's pretty obvious...