MMA is the type of fighting, the UFC is the biggest "major" promotional company who puts on MMA events, there is also Cage Rage/Elite XC, Pride, (now owned by the UFC and no longer promoting) M-1, Adrenaline etc.
There are hundreds of smaller "feeder" organizations that people can cut their teeth on fighting at the small shows before making it big and getting a UFC (or other big org.) contract.
Boxing and MMA are COMPLETLY different animals, you really can't compare the two. I am a much bigger fan of MMA, only a few boxers have ever excited me. And with only 2 (possibly) mma-related