Yes, I've read the developer documentation and even went through a few tutorials... but I'm still not totally clear on what I can do with Facebook Connect.
What I am guissing is that I can either have a user sign up and create an account on MY website, or the user can sign in with Facebook Connect and I can store their profile information in MY database.
Am I wrong?
What you can do with Facebook Connect is a very open ended question (ie lots of things). You should try to narrow it down a bit.
As for storing Facebook user profile information in your database, Facebook defenitly doesn't want that. You can store some data for short time (like 24 hours) to improve your site's performance, but not permanently. And if user deauthorizes your app on Facebook site you are supposed to delete all user specific data you got from Facebook.
Creating your own account linked to Facebook profile should be ok.