It depends on the game.
If it's an RPG:
-I like it to have bold and vibrant graphics
-Distinctive characters
-A large assortments of unique weapons and armor. Not "we have over 400 weapons, but they consist of sword, sword+1, sword+2, etc."
-To accompany the weapons, armor, and other equipment, a good range of spells and abilities is a must. These skills had better be independent of each other, and by that I mean I don't want 3 different colored energy balls labeled "fireball" "ice bolt" and "crap toss"
-I also like to have very diverse and customizable classes. Each class had better be significantly different than any other class, and reflect that in the way you play the game with your chosen class.
-I like having my own character, not a predetermined main character who looks the same no matter what equipment you give him.
-Good character development. I should feel some sort of emotion for the main character, and ideally I should be able to empathize with him.
-With good character development comes good story. I want a purpose for what I'm doing please.
-It would be nice if the warrior class has skills other than "strong attack," "stronger attack," and "whirlwind attack."
-I want to be challenged. I want to have to think a little. Involve some stratagy in battles.
-Speaking of battles, real time is best.
I think that's about it for RPGs, now, on to FPSs...
In an FPS I like to see:
-Graphics aren't as important as a good frames-per-second. I would rather have buttery smooth animation than hyper-realistic graphics.
-A wide array of guns, hopefully all with fairly unique stats, but it's understandable if some are similar to each other.
-Customization. The more the better. I think the recent CoD games and Rainbow 6 are doing good work with the customization options for weapons, but in the future, I'd like to see more.
-Unlockables. I like being rewarded for playing a game I like, and what better way than to have some sweet stuff available for either playing long enough or performing well enough? (Sweet, not good. I don't think it's fair if people who have played longer/better are given guns better than what the noobs or newbies have to work with).
-Varied maps. I hate playing an FPS where you're either playing in the jungle map, that one map with a barn, or the other map with a barn.
-Equipment. Guns are awesome and all, and they should of course be the main focus, but being able to equip yourself in whatever armors and gadgets is something I appreciate.
-I am one of those people who will tinker around with customization options that have little to no effect on game-play. So tons of pointless stats and knick-knack type add-ons are always fun for me.
-Fun game-play types. CTF and deathmatch of course, but the occasional bit of innovation never hurts. If it turns out to be a flop, there is always deathmatch.
-Emphasis on team play. I like playing a game where the only time you can take on 4 enemies at once is when you are a real BA.
-Intelligence is rewarded. Nothing irks me more than someone taping down the trigger and equipping an LMG. Being sneaky and/or smart in the way you attack the enemy team should net you far more kills than raining bullets down on enemies. Bullet-rain guns should be included, but, their focus should be on covering fire. Not as an assault weapon. Leave that to the rifles and SMGs.
Okay, wow, that turned out to be pretty long. I was going to put RTS also but...this post is long enough as it is. If anyone is curious what I like to see in an RTS just post, otherwise I'm done.