Nice choices.

for me it would be

Main character: Naked snake MGS3. For once the main character gets a real beating several times and still come off bad ass. Even though he's a soldier they give him heart.

Story: I agree heavy rain. The start to finish I was entranced in the story.

Gameplay: Maybe Battlefield 3.

Musical score: Any MGS, Harry Gregson-Williams is a genious.

Setting and Atmospere: Silent Hill 2.

Introduction: MGS2. Solid snake walking along th broklyn bridge in the rain then jumping off onto the boat is the intro that will stay with me for a long time

Ending: MGS4. People might say thsi was a cop out but one that ended like it should. Kojima did the right thing in my opinion.

Online: Also agree cod4 because at the time it defined a new age of online gaming

I think for me, I think new experiences and unique gameplay are when I have the most fun with games.

All of the above, especially originality, Ocarina of Time, Shadow of the Colossus, Heavy Raiin!

correct the title please
answer
Metroid Prime series is the perfect example
great visuals, tight controls, original concept, polished gameplay mechanincs, great progression, inspired ost...

Main character - Which main character is the coolest, nicest, cutest or just damn right brilliant?

- Conker!!

Story - Is there a story that made you cry? What about related to you somehow? Or still stays in your mind to this very day.

- The Legend of Zelda: Majora's Mask

Gameplay - The best game you've had the pleasure to control.

- Super Smash Bros series

Musical score - The game that contained the best musical score. Made you feel like a badass, soft and gooey or something like that.

- Pretty much any game made by Rare: Diddy Kong Racing, Golden Eye, Perfect Dark, Banjo-Kazooie series. All of them (plus more) nailed down the music perfectly on every stage. Robin Beinland is simply a genius.

Setting and atmosphere - What game made you feel like you was there? Made you tense? Gave you goosebumps?

- Halo: CE

Intro - The best intro to a game ever. So cool it would make Samuel L  Jackson kiss its feet.

- Donkey Kong 64

Ending - A game that wrapped things up nicely, jerked a tear or smile from ear to ear.

- Super Mario 64, The Legend of Zelda: Ocarina of Time/The Wind Waker, the actual ending to Conker's Bad Fur Day

Online modes - Something that's creative or an unoriginal online model but done very well.

- Halo 2

Still open for peoples opinions. The more the better.

Main character - Solid Snake (Metal Gear Solid 1), Cloud (Final Fantasy VII), Samus (any Metroid but Other M)

Story- Final Fantasy VI, Final Fantays VII, Final Fantasy X, Metal Gear Solid, Metal Gear Solid 3: Snake Eater, StarCraft (Brood War)

Gameplay - Mario platformers (2D and 3D), Mega Man games, Resident Evil 4, Mario Kart games (I could go on and on in this category since it's the most important of them all)

Musical score - Final Fantasy games (except XIII and XIII-2 which have horrible music), Secret of Mana, Donkey Kong Country games, Chrono Trigger, Mario Galaxy

Setting and atmosphere - Resident Evil 2, Silent Hill, Diablo I and II

Intro - Final Fantasy VIII (Liberi Fatali!!!), Super Metroid (The last Metroid is in captivity, the galaxy is at peace)

Ending - Final Fantasy X, Metal Gear Solid 3, Chrono Trigger (because there's so many!)

Online modes - StarCraft, Diablo, Mario Kart Wii, Super Smash Bros. Brawl (sue me)

I'll post reasons for some categories later.

Begin the title with Final Fantasy and end it with VII Remake.

Main character -  Link.  Archetypal hero, not super character driven, just a classic hero on a quest to save the world.

Story - Earthbound.  It hit every emotional note a game can hit.  It was sad, funny, poignant, and uplifting, often all at the same time.

Gameplay - Super Mario Sunshine.  Just absurdly fun, simple but innovative all at once.

Musical score - Probably Final Fantasy 6.

Setting and atmosphere - Flower.  I know this isn't what you meant, but the environments were perfect and made it possible to basically turn your brain off while playing and have a totally visceral gaming experience.

Intro - Blue Dragon.  From the way the splash screen pops up as you run down the hill, to the awesome, "LAND SHARK!  I WILL NEVER GIVE UP!" sequence, the first 10 or so minutes of this game are 100% unmatched ever.

Ending - Fable 2, to me, had the most awesome ending in history.  Never mind the DLC that kind of messed things up, the entire childhood dream sequence and the moral choice that followed it were very, very good to me.

Online modes - This is a hard one.  Timesplitters 2 was probably my favorite online gameplay experience in terms of pure fun, so I guess I will go with that?

Main character - A character that is very believable, but has a lot to learn. One that has strong morals and would fight for them. - Cole from inFamous

Story- A story should be strong and interesting. The characters have to be just as interesting it should cover themes make sense. -inFamous

Gameplay - Fun to play. Awesome to watch. -inFamous

Setting and atmosphere - The setting should be able to tell the story all on its own. -inFamous

Intro - Leaves many questions to be asked. Opens up with a bang to through plot. -inFamous

Ending - No plot holes, strong message. Most important of all, cannot be predictable. If i call someones death beforehand, and they do die, you are doing it wrong. -inFamous

Online modes - Simply fun and tactically balanced is enough. Not necessary though. Unless its obvious the game would benefit from it.