I'll break up what I love in 3 groups:
Fun : games where the gameplay is just fun, it could have bad graphics, no story and I won't care. This includes Wii Sports, SSX, 1080, Burnout(crash mode) Super Smash Bros, Power Stone, Crazy Taxi, Jet Set Radio, Warioware, etc...
Addictive : Titles that have strategy, leveling and stat upgrade type stuff, and some form of collecting concept(weapons, pokemon and such). They can also have okay graphics and okay story, but those three elements most be well balanced and extensive. This includes FF Tactics, Ogre Battle, Pokemon, Jewel Summoner, Sims, etc...
Atmosphere: This sounds like a weird cateogory but in essence it's game we're I just enjoy being in that gaming world. This includes any 3d Zelda title, Shenmue, Outcast, Dreamfalls, Jade Empire, Hotel Dusk, and in the future will include Mass Effect, Alan Wake, Thesis, etc...
I know these three groups kind of cross various genres and such but I think you get my drift here, 99% of the games I get appeal to me on one of these levels.
I could say I'm a fanboy of [enter company here], but to be honest the bottom line is I go where these three kinds of games go. Also I'm a fanboy of cheat codes and saving anywhere. While that sounds silly I've literaly decided for or against purchasing based on those two issues.