I think it is necessary in order to get me motivated to do the next think. When I see a cutscene that adds to the story alot I feel like I made such a great acheivement. So if a game doesn't have a good story , but good gameplay I might play it, but not as much as a game with a good story. I also feel that the way this story is told is important. Some games fit a text box more than voice chat, and others are the opposite. How important is a story to you?

Depends on what it is.

Brawl, Carnival, Endless Ocean, Mario Kart? Not at all.

CoD4, BioShock, RE5: Very relevant.

FFXIII, LO, Mass Effect: Critical.

 QFT!

I couldn't say it any better. I do prefer games to have a good story, but not necessary for some games. 

Think: Zelda with out story line...

...WTF?

yeah it depends on the game, but cut scenes are a +

I think story is really important in adventure/rpg games, It does make me wonder whats going to happen next if the story is that good.

Depends on the game. If I'm playing Final Fantasy then its very important. If I'm playing Gears Of War then its not important at all.

A deep, complex story isn't really important to me. I just need a basic background story. This isn't to say that I hate deep stories. If they are in great. If not, then whatever. As long as the game is good, then I'm fine with whatever story they put in (deep or simple).

Order of importance in games:

1) Gameplay
2) Atmosphere
3) Graphics & sound
4) Story, characterization, and all that jazz