jQuery is a javascript library. It's especially useful because it uses CSS3 selectors (the query part) to find elements on the page, and so is quite powerful with minimal syntax.
AJAX stands for Asynchronous Javascript and Xml. Originally AJAX implementations used the XmlHttp ActiveX components Microsoft provided for web based Outlook access. Later browsers (FX, IE6 and above, everything else) included it as a built in component.
However XmlHttp methods that AJAX uses can fit above any HTTP call, and so many calls use JSON or HTML instead. So ajax now means any asynchronous javascript method that calls the server.
jQuery includes an Ajax library, as do most Javascript frameworks.