At the moment they mostly represent the post-count. (afaik those groups came as default)
I.e. there are no additional rights bound to them.
The exception is the newbie state, which limits a few things (such as editing the signature) that are often used by spammers.
There are also groups such as "moderator" or "admin".
And there are groups for language boards (currently only Arabic has such a group, others may follow). This can be joined by each user via their profile. So if you want to join the Arabic board....
Internally there are other groups, that moderators can use to limit rights, if and where that is necessary (rarely happens).