JSON is one thing only: it is a format for exchanging data. When you encode your data in JSON format, it can easily be decoded. Say you want to encode your personal information:
my name is Juan and I'm 31 years old
you would encode it in JSON format as:
that string is small, easily transmitted, and easily interpreted; it is also standardized, so anybody with a compliant JSON decoder would be able to get your
age from that string. JSON is used by many web services to exchange data.
Now AJAX is not a language, or a product, or a software, or a specification, or a standard. AJAX is simply a way to do things. There is no definition of AJAX. Instead, AJAX is a collection of techniques where a given view of an Internet resource (say, a particular Web Page) can interact with the server asynchronously. For example, I am typing this message in a Text Box, and then I will hit
POST YOUR ANSWER. The server will receive this text, and put it in the site, but it will not reload the page (only the relevant part where it actually puts my answer).
AJAX is important because it provides a look-and-feel similar to a desktop application (it would be weird if every time you clicked something in your desktop application the application would close and then open again, right?)
So, JSON is a well specified thing (a data format) and AJAX is not (it is a way to do things, but there is no clear definition). They are unrelated, but both of them are widely used in Internet services, so you see their name pop up closely very often.