Each Rank has 3 divisions (1, 2, 3, with 3 being the highest), except for Challenger.
For example: You will climb in the following way:
Iron 1 -> Iron 2 -> Iron 3 -> Bronze 1 -> etc...

To rank up, you need to play Duels. If you win, you earn 4 points, and if you lose, you lose 5 points.
In other words, if you lose, you lose more points than you gain, so you always need to have a positive win rate to climb.