XMLHttpRequest() is an object (a constructor function) that allows you to send HTTP requests from JavaScript. Historically, XHR (XMLHttpRequest) was introduced in IE and was implemented as an ActiveX object. Starting with IE7, it's a native browser object, the same way as it's in the other browsers. The common implementation of this object across browsers gave birth to the so-called Ajax applications, where it's 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. This way, you can build much more responsive and 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; however, it can do other stuff and be notified, through an event, when the response arrives.
JavaScript because it's obvious that XHR objects are created...