Backbone.Controller was renamed to Backbone.Router last year. I guess you were reading an old tutorial.
BackboneTutorials.com is a decent tutorial for getting started.
Build a small app as you learn and you'll know what Backbone does for you.
The best way to learn Backbone is to go through the annotated source code. Each and every line is commented and explained.
Use the dev version instead of the minified version when building your app. This helps in debugging.
If you are into reading books, check out Addy Osmani's Backbone Fundamentals, a free e-book. The book is awesome, but quite verbose. You can use it as a reference.
See this question on Quora for the definitive list of resources;
Don't worry too much about what's 'V' and what's 'C' and backbone's spin on MVC.
The framework does an excellent job of separating code that interacts with the server and the code that performs the core client logic. Templating and updation of DOM Elements is left to you. It's pretty minimalistic in that sense. Backbone provides Events that make the different parts interact with each other.
