MMR, or Match-Making Rating is how teams are matched in rated PvP. MMR is separate from rating, but both are altered when you win or lose against other rated teams. Explaining the difference between the two is a little tricky, but essentially, MMR is how your matches are found. MMR is linked to players and teams, so a player forming a new team will carry some of their MMR with them from previous PvP escapades, in order for them not to face far lower-rated players. Rating is also linked to both players and teams, but doesn't dictate who you face.
Rating is won and lost via winning and losing against teams with better and worse MMRs. Say you're at 1500 MMR and 1500 rating, and you face a team at 1600 MMR and 1600 rating, and you lose. You will only lose a small amount of rating, let's say 5, and they will only gain a small amount, again let's say 5. Both your MMRs will adjust similarly. Now say you beat them. They will lose a big chunk of rating, but likely not such a big chunk of MMR. You, equally, will gain a big chunk of rating, and depending on how many matches you've played, likely gain a reasonable amount of MMR. The MMR is saying "hey these guys beat a 1600 team. They're better than a 1500 MMR".