FPS (Single Player): CoD: Modern Warfare... I liked the campaign. But FPS games in general usually have bland campaigns. It's nothing too spectacular. But I enjoyed myself playing it.
FPS (Multiplayer): Probably Quake III or Soldier of Fortune.
RPG: Star Wars - Knights of the Old Republic II: The Sith Lords - a pretty long title. But it's the only game where you can see stuff you won't see in the Star Wars galaxy.
TBS: Medieval II. But the diplomacy system is wrecked so it kinda ruins the experience, but it's still fun. Campaign mode - I don't like the real time battles.
MMORPG: World of Warcraft... But it's no longer fun :/
Action/Hack-n-Slash: - - -
Fighting: - - -
Racing: Midtown Madness. If only Microsoft could strike a deal with GoG and release this one. So, many memories!
Platformer: - - -
RTS: Age of Empires II. It has everything: teams, map editor, easy to play and you know. Fun in general.
Adventure/Point-and-Click: - - -
Casual: Klondike (or solitaire in Windows)
Other: Browser game I suppose? Well, Runescape (as a kid!) but they overhauled everything and made it suck.

FPS (Single Player): Painkiller
FPS (Multiplayer): N/A
RPG: Dragon Age: Origins
TBS: Civ II
MMORPG: Lineage II
Action/Hack-n-Slash: Diablo 2
Fighting: N/A
Racing: Mario Kart
Platformer: Super Mario - All of them
RTS: Heroes of Might and Magic 2
Adventure/Point-and-Click: Fatty Bear's Birthday Surprise
Casual: Bejeweled / Bookworm

This is good fun.
I can't say for sure, but here's some gut reactions as my favs:

FPS (Single Player): Thief the Dark Project (if it counts -- if not, Bioshock)
FPS (Multiplayer): Battlefield Bad Company 2 (probably will soon be Battlefield 3 after Christmas)
RPG: Baldur's Gates, probably.
TBS: Civ4
MMORPG: Guild Wars
Action/Hack-n-Slash: Baldur's Gate Dark Alliance
Fighting: Mortal Combat 1
Racing: Mario Karts (may 64 for my fav that I've played)
Platformer: Trine
RTS: Rise of Nations
Adventure/Point-and-Click: Drawn: Dark Flight
Casual: Free Cell
Space Sim: TIE Fighter
Flight Sim: IL2 1946 (got this 10 years late on GOG just a little while ago. I love it.)

FEAR. Excellent AI, awesome particle effects that make combat feel real and good weapons. The levels can be iffy at times, but overall I like the "real world gone crazy" feel to them.

Runner up: Crysis Morrowind. One of the most intricately designed worlds of all time, good lore and setting, interesting main quest, lots of choices, stat-based combat that makes your role more important than your personal dexterity. It's pretty much the RPG defined.

Runner up: Deus Ex (P.S. these are my two favorite games of all time) Super Metroid. Most of my early days on NES/SNES are not filled with games that stand-up as an adult, but Super Metroid is an exception. It's well-crafted, haunting even, with a great feel and visuals. It was interactive space opera way before Mass Effect came around, and still impressive.

Runner up: Batman: Arkham Asylum Super Mario World. I don't play them anymore really, just nostalgia trips through roms, but this is by far the one I spent the most time on. Just perfectly executed, not much else to say.

Runner up: Super Castlevania IV Sam & Max Hit the Road. Funny as hell, still beautiful to look at today and filled with good puzzles that challenge but don't frustrate. I could play it everyday.

Runner up: King's Quest VI

I don't really play the other genres enough to have a favorite, if I play them at all. Honestly I more consider myself a shooter/RPG fan than a "gamer" in the general sense. I think gaming is way too wide a category to justify calling myself a "gamer."

never heard about this game. what is it?

FPS (Single Player): Bioshock
RPG: System Shock 2
TBS: Terror From the Deep.Technically turn-based tactics, but I'm trying to be less pedantic =(
Fighting: One Must Fall 2097
Platformer: Mirror's Edge
Adventure/Point-and-Click: Day of the Tentacle
Puzzle: Braid
Shmup: Tyrian
Action-Adventure: Link's Awakening

Yeah, I'm mostly just a FPS person with some interest in "quirky" games as well (which is probably the reason I like Morrowind). I don't know much about RPGs, RTS, TBS, Platformers, Fighters, Racing games, or even hack and slash :P

Good choice with Super Metroid, btw. That game is really surprisingly atmospheric and engaging.

EDIT: oh yeah, and FEAR is awesome as well. It has some issues, but Monolith really knows how to make combat visceral and gritty.

No, I have a preview copy :P

... but yes, I meant COD: MW.

FPS (SP): Unreal Gold with its great atmosphere and beauty; Clive Barker's Undying, also very atmospheric; Doom (Knee-Deep in the Dead); Duke Nukem 3D.
FPS (MP): Battlefield 1942 with Desert Combat mod
Third-person action-adventure: Severance: Blade of Darkness
Racing: FlatOut 2; Indianapolis 500: The Simulation (1989).
Platform: Duke Nukem (1991); Out of this World.
RTS: Command & Conquer: Red Alert. Takes itself a little seriously (compared to the sequels) and has a great feel to it.
Adventure/point and click: Day of the Tentacle; Under a Killing Moon; Pandora Directive.
Flight sim (modern fighters/"study"): Jane's F/A-18 and F-15; Falcon 4.0 BMS.
Flight (modern fighters/"survey"): Jane's Fighters Anthology; Lock-On: Modern Air Combat.
Flight sim (historic military): IL-2

Hard to pick exact favourites in each genre, I'm afraid.

FPS (Single Player): Doom II. Doom is just too awesome for me, still fun today, and it will ever be. I end playing Doom or Doom II at least once every year, seriously :)

FPS (Multiplayer): Halo 3. Best multiplayer experience on console, with really different weapons, many modes, etc. If I had to say a PC title, Quake III without any doubt.

RPG: Tales of Symphonia. Great graphics, great story, many secrets, many great bosses, awesome character development... Vesperia is awesome too, and Final Fantasy VI is the best turn based jrpg, but I just loved too much this game :P

TBS: Advance Wars Dual Strike without any doubt, although all three Advance Wars are too awesome. Everybody should try this games out, I´ve put more than 500 hours in the saga easily, and I think it could be many more.

MMORPG: Haven´t played any, afraid of getting too addicted :)

Action/Hack-n-Slash: Batman Arkham City. Awesome action game, seriously. Great fighting, great sneaking, awesome characters and bosses, awesome story... The Darkness & Metroid Prime would be some other choices too (I know, Metroid Prime would be a mix of action/platforming/shooter, but whatever).

Fighting: Street Fighter IV without any doubt. I didn´t try much Street Fighter games before, but I started to love and to learn this genre after this title. Soul Calibur was fun to mash buttons, but this is totally awesome to learn.

Racing: F-Zero GX. Best racing game for me ever. I love kart games too though.
Platformer: Sonic 3&K being a Sonic fan (1, 2, CD, Generations, both Mario Galaxys, Rayman 1/2/3, Earthworm Jim, Dynamite Headdy, Ristar, most Castlevanias, most Metroids etc are AAA titles for me too as a platforming lover though, too hard to decide).

RTS: Dungeon Keeper is awesome, I think it´s the RTS where I have put more hours. For playing with other people, I prefer Age of Empires 2 though.

Adventure/Point-and-Click: Sam & Max Hit the Road. I think no game will ever beat that one for me. Day of the Tentacle would be a close second, and The Neverhood has a spot in my heart too, but Sam & Max is the perfect title of the genre for me.

Casual: Plants VS Zombies is a game everybody must try. Lots of fun guarantied. It gets old after perfecting nearly everything though, not that replayable for me.

Other (Anything that doesn't comfortably fit in any above category): Puyo Puyo 20th Anniversary. Puyo Puyo is the best of the competitive puzzle games, better than Tetris, Columns, Bejeweled, Hexic etc. That game is pure crack.

:>

FPS (Single Player): Half-Life 2 + episodes
FPS (Multiplayer): Quake III Arena
RPG: Arcanum / Torment (planescape)
TBS: Shogun 2 / Civilization IV
MMORPG: World of Warcraft (enjoyed SWTOR beta more than WoW though)
Action:Crusader: No Remorse / Metal Gear Solid 3
Hack-n-Slash: Diablo II
Fighting: Blazblue
Racing: Mario Kart (don't really play any racing games)
Platformer: Another World
RTS: Anno series / Starcraft series
Adventure:Little Big Adventuer 2
Point-and-Click: Broken Sword: TSotT / Under a Killing Moon / Day of the Tentacle
Casual: :( Solitaire maybe
Other: JRPG: Final Fantasy IX