I know that the Google Drive API allows for drive integration with Chrome Apps, but what about extensions? Can/How do I use the Drive API in a chrome extension? For example I want an extension that will have a popup that list a user's docs, how do I get access to their drive box? I know it starts with authentication, but I'm not sure how to do that either.
Regarding authentication, chrome.identity is the place to start.
Taking a (not so) quick look at the GDrive App I didn't really spot any app-specific stuff (i.e. something not available to extensions). So, I would certainly start by trying to "port" the GDrive App to an extension.
There is, also, this tutorial I totally recommend about Building Apps with AngularJS, which actually walks you through building the GDrive App. It is super cool, has a fairly detailed explanation regarding how to set up authentication related stuff (of course, you'll have to slightly adapt that to apply to an extension) and you can just ignore the AngularJS-related stuff (if you are not interested - although AngularJS is super cool too).
Recommended road-map:
[Check out Zig Mandel's answer below for an interesting alternative.]