Stars are rating points and you collect them to achieve the next rating title. After each game you will win or lose stars. The number of stars you need in order to achieve the next rating title will always be visible to you.
Once you won the title, you will never lose it, but losing stars will make you far away from the next title.
Also, the Division you will compete in in the weekly tournament depends on how many stars do you have. If you are in higher Division, you have a chance to receive better rewards at the end of weekly tournament.
Your position in the Leaderboard depends on number of stars you have in your possession.