What are your favorite games (no more than two) in each of the following genres? I'll list mine as well:

Platformer:
Super Mario World (SNES) / New Super Mario Bros. (NDS)

Racing:
Super Mario Kart (SNES)

RPG:
Mario & Luigi: Superstar Saga (GBA)

Sports:
Grand Slam Tennis (Wii) / Super Tennis (SNES)

Puzzle:
Tetris DX (GBC)

Adventure:
The Legend of Zelda: Link's Awakening (GB/GBC)

Shooter:
Contra (NES)

Fighting:
Super Smash Bros. Melee (GC) / Punch-Out!! (Wii)

Strategy:
Age of Empires II: Age of Kings (PC)

Let's see...

Platformer: Super Mario World (SNES) / Donkey Kong Country (SNES)
RPG: Chrono Trigger (SNES) / Golden Sun (GBA)
Strategy: Fire Emblem: Path of Radiance (GCN) / Advance Wars: Dual Strike (DS)
Adventure: The Legend of Zelda: Ocarina of Time (N64) / Metroid Prime (GCN)
Fighting: Street Fighter II (SNES) / SoulCalibur II (GCN)
Racing: F-Zero GX (GCN) / Mario Kart (DS)
Sports: Mario Tennis (N64) / Punch-Out!! (Wii)
Shooter: Sin and Punishment (N64) / Contra (NES)-Thanks pixelman for reminding me about Contra.

Some of these games don't fit perfectly into any of the categories, so I put them into the spot that made the most sense (to me).

Easy list!

Platformer: Crash Bandicoot (PS)
RPG: Record of Lodoss War (Dreamcast)
Strategy: Rise of Legends (PC)
Adventure: Fable 2 (XBOX360)
Fighting: It's a toss up between Soul Calibur 4 (XBOX360) and MK vs DC Universe (XBOX360)
Racing: Sega Rally CE (Dreamcast)
Sports: Any of EA's FIFA or NHL games (Multiplat)
Shooter: BIOSHOCK (XBOX360)

Note:
If any of you have NOT played Bioshock yet and you have X360 or PS3, I recommend you pick it up before the sequel comes out. It is possibly THE BEST FPS EVER! I've played through it various times and the story and the action never gets old.

@BrazCanaMan

Looking over your list, my best guess would be that you are either a teenager, or you haven't played many of the NES, SNES, MS, Genesis and TG-16 classics. ALthough I appreciate that your list doesn't necessarily mimic "the ol' standards" I have a hard time seeing Crash Bandicoot (as good as that is) as a favourite platformer with no Mega Man, Mario, Bonk, Sonic, Castlevania or any other super fantastically amazing platformer(s). Same thing goes with the fighting genre IMO as MK vs DC was marginally good at best but Soul Calibur 4 is a decent game. I would have at least excepted to see a KI, Street Fighter, TMNT: Tournament Fighter, Marvel vs Capcom, Samurai Showdown, Virtua Fighter or Tekken before MK vs DC. Anyway, I guess I shouldn't complain and compliment you are the same time. I'll just stick to complimenting your list's variety.

Platformer: Mega Man 2 (NES)
RPG: Final Fantasy VI (SNES)
Strategy: Ogrebattle 64 (N64)
Adventure: Metroid Prime 2 (GCN) or Castlevania: Symphony of the Night (PSX)
Fighting: X-Men vs Street Fighter (PSX)
Racing: Mario Kart Wii (Wii)
Sports: Ice Hockey (NES)
FPS: Call of Duty: World at War (Wii)
Shooter: Lords of Thunder (TG-16)
Puzzle: Dr. Mario 64 (N64)
Beat-Em-Up: TMNT: Turtles in Time (SNES)

Platformer:Mega Man 2 (NES)
Racing: Super Mario Kart (SNES)
RPG: Final Fantasy VI (GBA, PS)
Sports: None (Never liked Sport games)
Puzzle: Tetris
Adventure: The Legend of Zelda: A Link to the Past (SNES)
Shooter: Left 4 Dead (Xbox 360)
Fighting: Tekken 2 (PS)
Strategy: Dungeon Keeper 2 (PC)

@Maniac
I'm not teenager, I'm 28 =P. I started gaming since... well, since I can remember. I had (in no particular order) an Atari 2600, NES, TG-16, SNES, PS, DC, Saturn, Colecovision, PS2, GC, XBOX360, Wii, DS, GBA and of course PC's. My cousin had a Megadrive and Master system that I played for hours when I'd visit him in Brazil. So I know all the classics. My list is like what you said, I strayed from the path, I didn't want to list off the more obvious games that rule each genre. Also, I listed Crash Bandicoot because I never had an N64 (I didn't like Nintendo's choice to stick with cartridge. Plus Resident Evil was coming out for PS) and CB was the first 3D platform game I played and it kinda just fell in to the list because it was a memoribal game for me. In terms of fighting games, I do prefer 2D over 3D anyday, I just thought that those two are a little underappreciated (especially MK vs DC U). If you want my real "obvious choices" list, here goes:
Platformer: MegaMan 2 (NES)
RPG: Final Fantasy VII (PS)
Strategy: Shining Force (MD)
Adventure: Super Metroid (SNES)
Fighting: Samurai Shodown IV (Arcade / Wii)
Racing: Sega Rally CE (DC, I love this racing game, it can't be helped)
Sports: Hit The Ice (NES / TG-16)
I will break shooter down a bit to two parts. You can't compare FPS's to SHMUPS.
FPS: Bioshock (XBOX360, if you played it, you would understand it's greatness)
SHMUP: Soldier Blade (TG-16)

EDIT:
What? we got puzzlers now?!? OK
Puzzle: Columns (MD)

Platformer:
Ninja Gaiden (NES) / DK Country series (SNES)
Racing:
Mario Kart Double Dash (GC)
RPG:
Chrono Trigger (SNES)
Sports:
Mario Strikers Charged (Wii)
Puzzle:
Tetris Attack (SNES)
Adventure:
Zelda: Ocarina of Time (N64)
Shooter:
Ikaruga (GC)
Fighting:
Killer Instinct series (SNES/N64) / Super Smash Bros series (N64/GC/Wii)
Strategy:
Pikmin (GC)

Platformer:
Super Mario series (Super Mario Galaxy is my fave!)
Racing:
Mario Kart Wii
RPG:
Kingdom Hearts (PS2/DS)
Sports:
Hate them :/
Puzzle:
Tetris or the Art Style series
Adventure:
Legend of Zelda: Twilight Princess (Wii)
Shooter:
Metroid Prime 3: Corruption (Wii)
Fighting:
Super Smash Bros series (N64/Wii)
Strategy:
Hate them also (but i haven't played Pikmin so that may change!)

Platformer:
Super Mario World (SNES)

Racing:
Excitebots (Wii)

RPG:
Chrono T rigger (SNES)

Sports:
NBA Jam: TE (SNES)

Puzzle:
Picross DS (DS)

Adventure:
The Legend of Zelda: A Link to the Past (SNES)

Shooter:
Super Star Soldier (TG-16) / Blazing Lazers (TG-16)

Fighting:
Super Smash Bros. Brawl (Wii)

Strategy:
Don't play 'em. The closest would probably be Shining Force II

hmm let me think...

Platformer:
Super Mario 64
Racer:
F-Zero GX
RPG:
Final Fantasy: Crystal Chronicles (amazing with 4 players and 4 GBA's)
Sports:
Mario Strikers Charged
Puzzle:
Tetrisphere
Adventure:
The Legend of Zelda: Ocarina of Time
Shooter:
Jet Force Gemini
Fighter:
Soul Caliber II
Strategy:
Little King's Story

That was easier than I thought.

@BrazCanaMan
Ha ha, no need to "correct" your original list. I was just amazed by the fact that your oldest game was a PSX game. I was also surprised by the complete lack of Nintendo games on a site called "Nintendolife". You've got me interested in trying Bioshock one day though (despite not owning either system).

@Maniac
Heh, I didn't notice that there weren't any Nintendo games on my first list, that's funny.
Bioshock is available on PC as well. It's an old game so you can find it cheap. I HIGHLY recommend this game to any fan of the Metroid Prime series. I'm just psyched for the sequel that's coming out towards the end of the year.

Platformer: Super Mario World (SNES)
RPG: Final Fantasy VII (PS)
Adventure: The Legend of Zelda: Ocarina of Time (N64)
Fighting: Super Smash Bros. Melee (GCN)
Racing: Rush 2049 (N64)/Mario Kart 64(N64)
Sports: Mario Superstar Baseball (GCN)
Shooter: Perfect Dark (GCN)
Strategy: Starcraft (PC)

I don't exactly know where No More Heroes should fit, but that'd be in my favorites somewhere

Platformer: Any Super Mario; but I love Super Mario 64 more than Super Mario Galaxy... & Super Mario World is another favorite!

RPG: Any Pokemon game; I love all the versions... but G/S/C were the best! My favorite as of now are Diamond & Platinum!
A 2nd RPG is Secret of Mana!

Adventure: Any Zelda; I'm a Huge Zelda fan a million more times than Mario! Twilight Princess is my #1 Pick; but I love Ocarina of Time, Majora's Mask, A Link to the Past, & Wind Waker... those are my Top 5 Zelda's!

Fighting: Any Soul Calibur & Mortal Kombat! Soul Calibur 2 & Legends are my top 2 in that series; & MK is Armageddon & UMK for the DS! That's 2 for each isn't it?!
I also like the Smash Bros games; but after a while I get bored with them! My pick is Brawl though; but stopped playing it since I got done with the story & unlocking characters/trophies/etc!

Racing: Any Mario Kart hands down; favorite is 64 over Wii! But I love Excite Truck; since I love listening to my MP3's while racing!

Sports: Punch-Out Wii is loads of fun! Wii Sports; soon to be Wii Sports Resort... I've been done with the first since a long time!

Shooter: I bloody hate them; don't ask why... hate the genre completely & any violent games for that matter! Mortal Kombat doesn't count; I can stand that! And I can't even get into the Metroids half of the time! It's all of those freaking FPS; & I loathe people who like the stupid XBox360 & PS3!

Simulation: Endless Ocean & Nintendogs!

Platformer: Super Mario World (SNES)
Racing: Mario Kart Wii (Wii)
RPG: Dragon Warrior IV (NES)
Sports: Wii Sports bowling (Wii) but I haven't played any of the new WMP games yet
Puzzle: World of Goo (Wii)
Adventure: The Legend of Zelda: Majora's Mask (N64)
Shooter: Far Cry 2 (PS3) for single player, Halo (Xbox) for mutliplayer
Fighting: Super Street Fighter II: The New Challengers (SNES)
Strategy: Age of Empires II (PC)

Platformer
Super Mario World (SNES)

Racing
Mario Kart Wii

RPG
Does Starfy count? Cause if it does that's my favorite!

Sports
Super Mario Strikers (Gamecube)

Puzzle
None

Adventure
Legend of Zelda: Ocarina of Time (N64)

Shooter
None

Fighting

Super Smash Bros. Brawl

Strategy
None

Platformer:
Super Mario Bros, Super Mario Galaxy
Racer:
Mario Kart Series Burnout Legends (psp) - Excite truck rules too
RPG:
nope
Sports:
Fifa 07 to 09 Pro Evo Series on the Wii (notable mention for ace golf and Wii Sports)
Puzzle:
Tetris DS, Puzzle League DS
Adventure:
The Legend of Zelda, Wonder Boy 3 The Dragons Trap.
Shooter;
Golden Eye (FPS), 90's shmups in general (for shmups)
Fighter:
Soul Calibur II, Street fighter alpha 3 Smash Bros Brawl (I know thats 3 )
Strategy:
Sim City, Advanced Wars Dual Strike

Platformer:
New Super Mario Bros. (NDS)

Racing:
Super Mario Kart (SNES)

RPG:
ES3: Morrowind (Xbox)

Sports:
Madden 08 (PS2)

Puzzle:
Puzzlequest (NDS)

Adventure:
Fable (Xbox)

Shooter:
Blackthorne (SNES)

Fighting:
King of the Fighters 2000 (Arcade)

Strategy:
Empire Earth +expansion(PC)

Platformer:
Super Mario World (snes)

Racer:
Super Mario Kart: Double Dash (Gamecube)
Turn based RPG:
Crono Trigger or Phantasy Star 4
Action Rpg:
Y's Book 1&2

Strategy/Rpg:
Final Fantasy Tactics or Arc the Lad 2

Sports: Mutant League Football (genesis)

Puzzle:
Shanghai (Sms, and Atari Lynx)

Adventure:
Beyond Oasis (Genesis) or or Castlevania: Symphony of the Night (PSX)

Shooter:
Galaga (arcade) or Raiden (tg-16)

Beat-Em-Up::
Xmen: The Arcade Game (Arcade) or Double Dragon 2 : the Revenge (nes)

Run N Gun:
Gunstar Heroes ( Genesis)
Fighting:
Soulcalibur II ( Game Cube)

Strategy: (None)

Maze /Labyrinth - Pac-man (Collection (Atari 7800- It's a Homebrew game) or Jr. Pac-man (Atari 2600)