This is a complicated subject but here's my attempt to distill what all of these things are and how they relate.
AJAX: This is a concept that involves pulling data asynchronously into a web application without reloading the entire page. It allows for dynamic UIs such as Google Mail that act like desktop applications but run in a browser.
ASP.NET AJAX: This is a library authored by Microsoft that attempts to make the AJAX concept accessible to developers that have an ASP.NET background. It provides methods that are similar to .NET Framework methods such as String.Format and others. It has some overlap with JQuery but they can both be used in harmony together as they have different strengths and weaknesses. It is important to remember that this is a client-side library and doesn't actually have a dependency on ASP.NET. You could use ASP.NET AJAX with any server-side technology.
Now, as to the question of what you should learn as an ASP.NET developer. I would recommend getting familiar with the functionality of both ASP.NET AJAX and jQuery. Microsoft endorses both, and both are included with VS 2010. They have different strengths and weaknesses and if you understand what those are you will be more effective with using them.
You don't have to use AJAX to do ASP.NET websites, but your websites will be more powerful and more user-friendly if you do. And if you develop a sense for when AJAX is effective and when it isn't, you will be a better web developer.
Oh, and stay away from UpdatePanels. That was MS's first answer to AJAX, but they are generally bad and go against a lot of the concepts that make AJAX good. They give you a short term benefit but in the long term they are a maintainability nightmare in my experience.