Well, usuallly there is something like a judge or a group of judges. They basically set up the whole thing. First they open up auditions so that people can go create their character and send it to them. They select an amount of like 32 or 64 people and the tournament starts.
The battles are usually 1 versus 1. After a while the judges decide winners and the amount of competitors decreases (so if there were 64 people at start, only 32 continue to round 2). This continues until there is only one left, the winner. :3
I think it would be very cool to have some anthro/furry tournament