MMO stands for Massively Multiplayer Online game. It's a game played on your computer (or, in some rare cases, on a gaming console). The unique thing about a MMO is that it is "persistent"--that is, action continues to happen in the world even when you're not there playing. Unlike a regular computer game, where you only control 1 character & everything else is computer-controlled and -generated, in a MMO you'll be playing alongside other players from all over the world. It's not uncommon to be playing with people from different states, all at the same time--in fact, depending on what time of day you're playing, you could be playing with people from other countries! Another acronym for MMO is MMORPG, or Massively Multiplayer Online Role-Playing Game. Think of Dungeons & Dragons--only played online. The whole thing sounds very complicated, but really isn't.
One important thing to remember about playing a MMO is that you're not done when you purchase the software. The vast majority of them are subscription-based; you must pay a regular amount per month in order to access the game servers, without which you will be unable to play.