[@Kimchi] When you're level 30, you can play competitively, which in turn places you in ranks. There are five divisions in each league, 5 being the lowest and 1 being the highest.  

The leagues are:

Bronze
Silver
Gold
Platinum 
Diamond
Master 
Challenger 
 
You need 100 LP (League Points) to climb ranks. After you get 100 LP, you go into your promotion games, which is a set of 3 very important games. Win two of them, and you get into the next rank with 0 LP.

For example, I'm Bronze 4 with 79 LP. If I win my next ranked game, I'll be Bronze 4 with 100 LP, and I'll be in my promos. If I win 2 of the 3 promo games, I'll be Bronze 3 with 0 LP.