MMR, or matchmaking rating, is basically the old ELO system, but as a hidden thing in the game. It determines who you're placed against and how much LP you gain/lose per match. 1,200 equates to about silver 5. Your MMR relative to the enemy team's average MMR is what determines exactly how much LP You gain/lose. As an example, if you go up against enemies with 1150 average MMR, and win, you will gain more LP because the game didn't expect you to win. If you go against players with 950 average MMR, you'll gain less LP if you win than you would against higher MMR. You'll also lose more LP if you lost a game that the system expected you to win based on MMR and lose less if the system expected you to lose.
Also like the old ELO system, MMR doesn't have any kind of gateway between two tiers. You don't have to win a certain 2/3 games to go from bronze 2 to bronze 1 average MMR, but if you lose enough games in a row, you can drop.
Finally, MMR increases faster for win streaks and decreases faster for loss streaks as the game's way of saying, "You're not in the MMR range you should be, let's get you there quickly."