First of all, Backbone is cool; if you use it, by extension you are cool 8) !
Backbone is a framework that provides you with awesome tools to develop your web applications. From their site:
Backbone.js gives structure to web applications by providing models with key-value binding and custom events, collections with a rich API of enumerable functions, views with declarative event handling, and connects it all to your existing API over a RESTful JSON interface.
I like to think about Backbone as an “MVC” (without actually the “C”). In my words, Backbone is a collection of objects with functions that provides structure to your code and make your life way easier by providing you with awesome tools to work with your data, html, events, services, etc.
Backbone is not a replacement of jQuery, in fact it requires jQuery as a dependency.