Looks like you've answered your question partially. MVC is a software architectural pattern which is used in implementing User Interfaces. So this is basically used as a pattern which allows you to follow the framework and make it easily understandable. AJAX (Asynchronous Javascript and XML) is used on the client side to create Asynchronous Web Applications. Using this, you can make calls and get or send data from the client side on to the server side on the background. This helps in making the User Interface experience of the user much better.
So, for instance, let's say you make a login page with a username and password. After the user clicks "login", let's say it takes 2 minutes (I know, this is an extreme case) to actually login the user and make use of their credentials.
If you use the classic MVC, the user might not be able to do anything during these 2 minutes (Because you're doing all the data processing/data binding on the client side). But if you use AJAX, you can make all of the data processing in the background, and in these 2 minutes, you can make the user play a game/do something else. AJAX is mainly used in cases where you want to get some data from the server without refreshing the view (Remember, Asynchronous).