There are 2 types of clubs: Standard and Custom

For each show type there is a standard club. A horse can be member of no more than one standard club, and that membership will bring a certain bonus when the horse is entering shows of that type.

The custom clubs exist (right now) for the most popular breeds.
A horse can be a member of 5 custom clubs.
The clubs are meant to be gathering places for people who like the same things, so in the future, there will be custom clubs other than breed clubs.
Each custom club has a president who will be elected by the members of the club.
The president is in charge of accepting/rejecting new members, as well as other powers not currently available ;)
Anyway, each club has a message board, so feel free to try joining them.