There are 7 leagues in total. These are, from the highest to lowest:
After each race season ends, the clubs with the highest scores go to the higher league (if you're already in the highest league, you stay there) and gets the best rewards available. The 3 clubs with the lowest scores (if you're already in the lowest league, you stay there) don't get rewards and go to the lower league.
You can check the leaderboard to find out if your club will go down or go up. When you see a green arrow next to your club, you go up to a higher league. When you see a red arrow, you go down to a lower league and you do not get any rewards.
If you don't see any arrows next to your club, you stay in the same league and get rewards.