There's a competitive element to nearly every game in existence but MMOs always seem to bring out that side a touch more than other genres. Obviously, most MMOs feature a Player Vs Player mode of some sort but World Of Warcraft has shown how competitive Player Vs Environment modes can be too. World Of Warcraft's endgame is filled with raids and instances and whenever a new one is released, guilds across the world race to finish the content first. Some guilds put in day-long shifts to learn boss tactics and gear their characters perfectly to give them as much chance at clearing the content as possible. Blizzard encourage this too - the in-game achievement system rewards the guild who clear a raid first on their server with a unique achievement to show off (there's also a message which gets sent out across the server to indicate a group has cleared an instance first). While very few of us will get to experience the highs of clearing a raid first, it's interesting seeing the extremes players go to to best rival guilds. Of course, for the rest, we have PvP modes. World of Warcraft does these better than most MMOs too with a range of game types and maps. Some have been around for years now and they're every bit as exciting as they were back in the day: Warsong Gulch - a Capture The Flag map - is still an intense boxed battlefield where defence is every bit as thrilling as attack. The Arena system implemented in The Burning Crusade expansion is also highly popular with smaller groups (2v2, 3v3 and 5v5) going against each other to advance up ladders. Again, us mere mortals are unlikely to ever bother the top of those ladders but it's still fun to grab some friends and try your luck in the arenas of Azeroth. For a further example of the highly competitive (and addictive) nature of World of Warcraft, the "Make Love Not Warcraft" episode of South Park is highly recommended.
I have an addiction to achievements, a craving for new bands and a dream to become Captain America. I once finished second in a Mario Kart 7 tournament so I'm kind of a big deal.