Tim, if you want to get into the business of working with Facebook, you need to know right away that they are always in the process of updating. That being said, you should learn some Facebook terminology so you can do more research and figure out what you want to do.
Applications that run within the Facebook interface are called Canvas Apps. Your application is hosted on your own server, and Facebook simply adds an iframe in their UI that points to your app's URL.
Another way to go to create a webapp that uses your own design, but still utilizes a single sign-on system from Facebook. This used to be called a Connect App but now they don't really have a name for it. Basically, you can use Facebook's authorization system to pull Facebook user data into your own database, which basically eliminates the need for a registration page. The authorization system prompts the user to allow your app to get specific data about them - once they authorize it, your app has access to do whatever the user gave it permission to do.
The best document to wrap your brain around is about authentication: http://developers.facebook.com/docs/authentication/
Give that a whirl, and good luck!