Recently I have been working on Ajax. So according to me AJAX displays content in HTML using XML. But now does this mean it is a Rest api.
Using REST we can do operations (PUT,POST,GET,HEAD) but by using AJAX we can only retrieve data from server side , AJAX can be a part of REST but REST can never be AJAX http://rest.elkstein.org/2008/02/ajax-and-rest.html
AJAX is a set of (typically) client-sided web development techniques, while REST is an architecture style for sending and handling HTTP requests. So you can use AJAX to send RESTful requests. A REST API is typically not implemented using AJAX, but can be accessed by an AJAX client.
There is plenty of information on both AJAX and REST (API) on the Internet. It should be easy to find.