There are currently five leagues in the game: Bronze, Silver, Gold, Platinum and Diamond. The key difference is the number of tasks to complete and the reward for winning. The Bronze League allows you to complete 5 tasks + 1 task that can be bought with farm money, the Silver League allows you to complete 6 + 1, the Gold League allows you to complete 7 + 1 tasks, Platinum - 8+1 and Diamond - 9 +1.
The clubs that finish the race in the top three advance to the next league. The only exception is the Diamond league, you will stay there if you win the race.
Clubs finishing in the last three places will be promoted to the previous league (but no further than the bronze league).