Hmmmm. Well, co-op is big for me. I am a social creature. I want to play my games with people. If it has a co-op story or co-op play, then I will likely buy the game. Borderlands is a game I bought all the DLC for. Love that kind of co-op.
However, I really don't care for vs. maps. It just isn't my thing. I played sports in high school, but quite honestly, I preferred the scrimmage games that we did in practices more than the actual matches. Yes, for me it really isn't about who wins or loses, it's about fucking playing. I love playing and having fun. Vs. to me is all about gloating. "I beat you!" followed by the Nelson "HA HA" is just not my idea of fun. If you like, you can believe it's because I suck at it. But I know I will play a game I like until I'm good at it. And since I don't like vs. maps, I'll probably never be very good at it.
I also enjoy strategy games. Things where it's not just about sheer reaction speed, but about planning as well. I love RTS games of any difficulty. As I've recently discovered though, they really have to be played on PC.
And for the games I play solo:
Sandbox games - I like choosing what to do next, rather than being put on a linear path. Fallout stands out for me, for many reasons, but put post-apocalyptic in your setting, and I am buying that game. I love post-apocalyptic stuff. I've logged more hours into Fallout then I care to admit.
Platformers - Mario, Ratchet, Sly, Sackboy... making a difficult jump and snagging that rare collectible, I kind of love that.
Racing - I do enjoy racing against others. But career mode is where it's at. Buying cars I could never afford, then driving the shit out of them... that is awesome.
That probably mostly covers it. My gamerscore in both PSN and XBL give a pretty good picture of what I like to play. If I have more than 50% of the trophies, I probably love that game.