How does the division system work?

First you have to play some games in the corresponding matchmaking queue to get placed in a division. In the leaderboard tab you will see how many games you have already played and how many you need to get placed.
Each division corresponds to a rating range. The system will then put you in a fitting division. More precisely, it will put you a bit lower than what would fit most to your rating. This acts as a soft reset every season. Your starting points will also correspond to your rating, although this is only really relevant in the highest division.
Once you are in a divison, you will earn points for a win and lose points for a loss. By default this is 1. While you are lower than your intended division, you will earn 2 points for each win. If you should ever be in a higher division than your rating suggests, you will lose 2 points for every loss. Keep in mind that your underlying rating can also change. This may lead to the boost starting or stopping even if your division has not changed. In the highest division you will even get a boost if the system deems your score to be lower than it should be.
You will be promoted when your score exceeds the score maximum of your division. You can gauge that by looking at how much the score circle is filled. Similarly, you will be demoted when you fall below zero points. You will get a certain number of points in the new division, so you don't fall back immediatly to your previous division.