Leagues are the tier system in campaign mode in which you battle through, conquer and are promoted into from the lower league.
The player starts off in the green league, the player fights through a collection of 5 trainers at a time called a league match. Every time the player beats a league match, the player gets an upgrade in their ?/? status (the number on the right represents the number of trainers in the league, the number on the left representing the position you are in the league). Once the trainer's left number is 8 or lower in their ?/? status (or they are one of the top 8 players in the league) they can compete in tournament mode. Tournaments are simple series' of matches with 8 players if you win you are able to take the promotion test.
In promotion test the trainer faces of against the trainer who runs the league. If you win, the trainer promotes you to the next league and ups your rank.