You play 10 placement matches. That determines your league (Copper, Bronze, Silver, Gold, Platinum). Once you are in a league, you are put into a division within that league (ie. Silver 31, Silver 34, etc.). A division is just up to 100 people in the same league as you for you to compare your leaderboard ranking. The idea is that if you're in a 100 person division, you'll feel more motivated to try to get to the top than if you're in a 1,000,000 person league leaderboard and won't care about moving up.
Once you're in a League, you can move up/down by winning/losing too much against people in the same League as yourself. For example, you goofed off in Placement and lose every single game, ending up in Copper, but you're way too good and bash all the noobs there, so the system will notice that and bump you up to Bronze. The question is how long it takes for the system to realize you're not in the correct League and move you.