Story tends to be important to me, so I'll go with that. It's not always important to me, mind you. It depends on the game in question. Sometimes the story's really simple, and the focus is on gameplay. In that case, the gameplay matters most to me, and the story matters little, if at all. However, I can say that the story has kind of saved a small handful of games for me. The ending I got in Silent Hill: Shattered Memories makes me glad I played it, though that doesn't necessarily mean that I'll play it again.
I can't say graphics don't matter to me at all. I play handheld games and retro games in addition to modern console games, but I do have a preconceived notion of what, say, a ps1 rpg out to look like. For the most part though, art style matters more to me than attempted realism. I'd rather have a game that's pleasing to look at, regardless of how detailed the graphics are. I'd honestly rather look at something like Chrono Trigger than CoD. Of course, that doesn't mean a game can't look pleasing, and impressive at the same time. Water in the Uncharted games is unnaturally blue, despite that series being among the best looking games ever released on a console. I like that design choice though, since that blue is easy on the eyes. For the most part though, I would rather play an enjoyable game that looks awful than a good looking game that's unpleasant. That's just me though.