Story is pretty important to me. In fact, that's what I value most in the game now if I want to finish them. Of course, gameplay is still more important but the story should complement with it very well.
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I have a difficult time listing aspects of games by importance. I suppose it would really depend. I've been playing Spec Ops for a bit, and I normally don't play military shooters, but the concept behind the story is what gets me into it.
Other games I don't follow the story at all because the events and concepts are told in a way that doesn't interest me or the concept itself doesn't interest me, but I play anyway, ala Demons Souls.
Then there are games that seem naked without their story and some that have amazing stories that aren't really required to be good.
It needs a perfect balance between the two in single-player focused games. If I can't get into the gameplay, I will likely not want to play. On the flip side, if the story is putting me to sleep, it will likely not last me long enough to finish.
PSN ID: Troll_Face_Flame (formerly armyflame) X360 gamertag: ArmyFlame9 The more people post on GameFaqs, the more I lose faith in humanity.