Inviting friends is something that Facebook does offer. Basically, it invites them to join the Application as well. “Community” isn’t a FB term that I know of.
When people connect to your site via Facebook Connect, they’re granting your Application specific permissions. Namely, the ability to get their user information and such. That permission granting makes them a user of your Application.
Inviting their friends is a Facebook specific thing that can be implemented (rather easily, I think). This just lets them send messages to their friends saying “hey, come check out this Application”, basically. Not sure it makes sense in terms of a site’s Facebook Connect application, as really it’d be people inviting their friends to come check out your website. I’ve have to test the idea, see how it works.
Anyway, when a user connects to your site, you get their Facebook user profile number. By keeping track of these, you can determine who is using your application. So that’s what that plugin means by “community”: The total list of users that have ever “connected” to your site.
The way it does it is by creating additional WordPress users for every new Facebook user. I specifically did not want that, so my plugin does not keep track of these numbers. I could keep track of them and show that community box, but I haven’t yet. Nobody’s asked for it. Personally, I like the idea of Fans better. But that’s me. ??
My main focus has been for a single-user blog type site, I haven’t considered multiple WP users much as of yet.
FB Groups are wholly different things, entirely within FB itself. Normal FB users can join groups and post messages and such in those groups.