I don't mind when a game doesn't have any sort of story, or only has one or two 10-second cutscenes.

I also don't mind when a game has a moderate amount of cutscenes. (Which to me is around one eleventh-twelfth of the total length.)

So what kind of a story do I prefer? A story, that is deep, intriguing and elaborate, but has as few cutscenes as possible. A story, that the player can skip, without being punished by having no idea what the funk is going on, and why the player is doing what he/she is doing.

The best (and probably the only) example of a story like this, is Metroid Prime.

Yes your right, every game does have a story. For the sake of the thread, no story are games that only have an action/reaction type of story. You know like taking something and then asking the main character to get it back without further development on either the story or the emotional state of the character. Something along those line.

Metroid prime seems to do well in this regard, it was already chosen by a couple of posters.

As long as it isn't in the way, well most of the games anyway.

If a story is well written, like say Okami or Ocarina of Time, then yes. 

The problem is that most of the stories in video games range from mediocre to awful.  Normally I like little to no story because most of them are badly written.  If it's between a bad story or no story, I'll take nothing. 

Also, some games don't need a story.  Remember when they tried to write one for Monkey Ball?

Most of the time I like if the story is a major part of the game. But in games like tetris or Plants vs. Zombies... well if its funny, then Im sold, if there isnt any... I dont care

The story is the most important aspect of a game IMO. I need to feel that their is a reason/purpose in what i'm doing. It is the only aspect that I would put above Gameplay. This is generally why I probably why I like the JRPG genre so much.

Depends...some games are just meant to be arcadey, and in those, like DMC and Ninja Gaiden, I don't pay attention to the story whatsoever.

Now, on the other hand, RPG's and blockbuster titles usually require a good story.

I love it when a game has a great story, i like to get the same experiance as you do when reading a book, when you get immersed in the story and like to know whats going to happen next.I know its not to everyones taste but to me the games that have good stories are easily by a long way the best games to play

Just look at the all time great games like, Metal Gear Solid series, Shenmue series, Final Fantasy series, Mass Effect series, Legend of Zelda: Ocarina of time, Legend of Zelda: link to the past, World of warcraft, Super mario series, Resident Evil series. all these games have storylines and will almost always be in a top 10 list of greatest games.

so id say having a good engaging storyline helps massively.

Depends on the genre. Certain game genres (mainly traditional core genres) and storytelling tend to go well together. RPGs and adventure games immediately come to mind (the heavy story option). But other less obvious genres like FPS/TPS, RTS, platformers and even sports games (where you start your own character/career) can benefit from some sort of narrative (the light option). In these genres the narrative isn't essential however, but is nice to have something present to push the game along.

Other genres however don;t need any storyline whatsoever and can thrive completely on gameplay like puzzle games, beat-em ups etc.