An Ajax call is an asynchronous request initiated by the browser that does not directly result in a page transition. A servlet request is a Java-specifc term (servlets are a Java specification) for servicing an HTTP request that could get a simple GET or POST (etc) or an Ajax request.
A good example of an Ajax request is the comment system on Stackoverflow. You can enter a comment in the textbox and click submit. It doesn’t submit the whole page (like a traditional HTML form submission would, which translates into usually a POST but sometimes a GET HTTP request). Instead the browser will send probably a POST request via XHR to the server and be notified of the response (hence “asynchronous”). But the server typically can’t distinguish between an Ajax request or a page transition because both simply come down to HTTP requests.