The only similarity between AJAX and Node.js is that they are both JavaScript; yet they both serve completely different purposes.
AJAX (short for Asynchronous Javascript and XML) is a client-side technology, often used for updating the contents of a page without refreshing it. Such use can be seen just about everywhere; from Facebook, even to Stack Overflow.
Node.js is server-side JavaScript, used for developing server software. For example, you wouldn't want to develop a database system in client-side JS (since, well, anyone could access the database's credentials obviously), but such a task is possible with Node.js. Node.js doesn't get executed in a browser, but by a server - it's uses are very comparable to Apache/PHP, Ruby on Rails, Django, and the like.
TL;DR: AJAX is for client-side stuff, Node.js for server-side stuff
