It's like being someones "friend" on Facebook, it kinda has that connotation.
In terms of what it actually "does", well not as much. You can adjust privacy settings for your profile, and one of the options is people you follow. That can be handy, also it's easier to see if someone you follow is online.
There is a modification to this forum software (that we haven't got, I think it costs money) that alerts you when someone you follow starts a thread, effectively making it easier to see what your "friends" are doing at CF.