For me, it's in this order:
Gameplay > Art Direction > Music > Graphics > Story.
The most important aspect of a game? Being able to play it, for one thing. That's why gameplay's up at the top. It doesn't have to reinvent the wheel, but if it's fun I'll give it a try.
I appreciate art direction more than anything else. If a game has the right art style and atmosphere, it all comes together much better than having HD graphics but with varying artistic styles. A single, coherent style that expresses the game's ideas properly is necessary. I don't want rainbows popping out of monsters when I kill them in a game that's supposed to be dark and scary. ...Or do I...?
Music I put in the middle, because repetitive or boring music gets on my nerves FAST. It should shine but not overpower, that is unless the game's about music.
I'm not looking for graphics, but it's hard to knock it down completely IMO. A game can have a good artstyle, but depending on how well the game utilizes the platform, it can go either way. Some game ideas are better with high definition, some a little less. But if it's fun and appealing to the eye, I'm not one to care what the resolution is or all of that other stuff.
I don't care about story. A lot of games I play are fine without plot explanations or cutscenes (Doom is my primary example). Some games have simple stories, mostly older games like Super Mario Bros. and Doom. That's all they need, really, and I'm okay with that. I've played games where story does matter (aka Phoenix Wright and Monkey Island), but it's usually not what I'm after in, say, action games. All I'm saying is that whether or not the story's there, it doesn't matter to me.