I think when it comes to things I look for in a game, it goes like this, from most to least important:
1. Fun - this comes from the controls, how responsive they are, and the overall gameplay. The joy you get from actually controlling the action onscreen. The less control I have over a game, the less fun it is.
2. Sound and music - Yes, even more than graphics or visual style. The music and sound effects of a game leave a much longer lasting impression on me than visuals, and enhance my overall experience much more.
3. Challenge - Very important. I don't want to just plow through a game. Most of my fondest gaming memories come from me overcoming a really tough obstacle or a difficult puzzle in a game.
4. Graphics - art and or tech - important, but art and style are more important than technology. As long as the graphics convey what they're going for, I don't need modern tech or HD or whatever else to enjoy it.
5. Story - not important. In fact, a lot of the time, I could do without it altogether.