Well its simple really, just say which game you feel was the best of the listed categories.

Most Played:
Super Smash Bros: Melee

Best Looking Game:
The Legend of Zelda: Wind Waker

Best Multiplayer:
Super Smash Bors Melee

Best Single Player
The Legend of Zelda: Ocarina of Time- Master Quest

Best Puzzle:
Super Monkey Ball 2

Best Original Game:
Luigis Mansion

Best Racer:
F-Zero

Best Shooter:
Metroid Prime

Best RPG:
Tales of Symphonia

Best Sports
SSX3

Biggest Dissapointment:
SSX4

Best Game
Super Smash Bros; Melee

Do feel free to add more categories

Most Played:
Super Smash Bros: Melee

Best Looking Game:
Resident Evil 4

Best Multiplayer:
Super Smash Bors Melee

Best Single Player
Resident Evil 4 or Metal Gear Solid

Best Puzzle:
Don't think I own a puzzle game ;]

Best Original Game:
Luigis Mansion.

Best Racer:
Mario Kart. F-Zero looked good, but I didn't enjoy it half as much as Mario Kart

Best Shooter:
Metroid Prime. Goes without saying. MP2 wasn't as good.

Best RPG:
Zelda: Wind Waker. I aint an RPG fan.

Biggest Dissapointment:
Probably Zelda. It was too short, lacked dungeons. There wasn't a wide enough variety of enemies. There seemed to be the same 10 all through the game. Sailing sucked really badly. Don't get me wrong, I LOVED the cell shading. But the dungeon boss's didn't look as mean as they could've done in more realistic graphics.

Best Game
Now this one is really hard. I'd rate this one on the game I had the most fun on. Working on that basis it is probably Smash Brothers Melee. I think worthy notes go to the whole Resident Evil series and Metal Gear Solid Twin Snakes!

Hmm..

Most Played:
Metroid Prime

Best Looking Game:
The Legend of Zelda: Wind Waker

Best Multiplayer:
Mario Kart: Double Dash!!

Best Single Player:
Metroid Prime

Best Puzzle:
Super Monkey Ball

Best Original Game:
Luigis Mansion

Best Racer:
F-Zero GX

Best Shooter:
Starfox Assault

Best RPG:
The Legend of Zelda: The Wind Waker

Biggest Dissapointment:
F-Zero GX

Worst Game:
Shadow The Hedgehog

Best Sports:
SSX Tricky

Best Game:
Metroid Prime

This probably looks similar to everyone elses with the lack of games for the cube, but here goes..

Most Played:
Mario Kart Double Dash

Best Looking Game:
Geist

Best Multiplayer:
Zelda Four Swords

Best Single Player:
Metroid Prime

Best Puzzle:
Zoocube

Best Original Game:
Pikmin

Best Racer:
F-Zero GX

Best Shooter:
Ikaruga (a proper shooter)

Best RPG:
Zelda Wind Waker

Biggest Disappointment:
SSBM

Worst Game:
Animal Crossing

Best Sports:
Beach Spikers

Best Game:
Mario Sunshine

Most Played:
Super Smash Bros: Melee

Best Looking Game:
Fzero GX

Best Multiplayer:
Super Smash Bors Melee

Best Single Player
Eternal Darkness

Best Puzzle:
Super Monkey Ball

Best Original Game:
Viewtiful Joe

Best Racer:
F-Zero GX

Best Shooter:
Metroid Prime

Best RPG:
Paper Mario 2

Biggest Dissapointment:
Crystal Chronicles

Best Game
Paper Mario 2

Worst Game
Die Hard Vendetta (a fooking disgrace is what it is)

Ok, here we go...

Most played: Smash Bros Melee - Love it. Can't get enough of it. Awesome game.

Best looking game: Zelda: Wind Waker - Perhaps a cliche answer, but the cel-shaded world is vastly enjoyable. Versatile and well plotted. One of the best looking games of its style ever.

Best multiplayer: Smash Bros Melee - Awesome MP action - suits heaps of styles and difficulties. I've had many a Smash Bros Session with my friends and even my parents. Unmissable, once again.

Best Singleplayer: Aww... I really wanna say the original Zelda: Ocarina of Time for N64, but on GameCube, Master Quest is a close second, so I'll say that.

Best Puzzle: Super Monkey Ball, the original. There's so much fun to be had, what with eternally plunging smiling simians to their untimely deaths. And then again. And again. Oh, right... there's a goal to this floor...

Best Original Game: Original? They're getting scarce nowadays, but I'm going with Super Monkey Ball again.

Best Racer: Mario Kart - Double Dash!!. A bit slippey-slidey at first, but there's an incredible amount of fun to be had.

Best Shooter: Metroid Prime 2 ALMOST as much fun as the original, but still as exciting as blasting space pirates right in the mandibles.

Best RPG: Pikmin - maybe this goes under 'puzzle', but anyways... I still love it.

Biggest Disappointment: Namco Museum ... i don't want to talk about it. $100 dollars and three hours later and it was still crap. Worst decision I ever made.

Best Game: Smash Bros Melee, end of story.

Worst Game: I've already answered this. Look two above.

Best Sports: Mario Smash Football So unrealistic but so good. I recommend.

And my own category...

Best remake: Super Mario Sunshine. So far, (apart from Mario 64), I reckon this is one of the best series remakes of mario's adventures. At least until Mario Galaxy.

Most Played:
Resident Evil 4

Best Looking Game:
Resident Evil 4

Best Multiplayer:
Monkey Ball party games

Best Single Player
Resident Evil 4

Best Puzzle:
Mr Driller

Best Original Game:
Pikmin

Best Racer:
There weren't many were there! I would say Burnout 2!

Best Shooter:
Metroid Prime (or Ikaruga if you mean shooter in the retro sense! hehe)

Best RPG:
Wind Waker - I don't have time for full blown RPGs!

Biggest Dissapointment:
Mario Sunshine. It just wasn't much fun! I am still traumatised by the horrible camera on the pirate ship!

Best Game
I found myself constantly coming back to Monkey Ball 1. I suppose I had the most enjoyment overall out of Resident Evil 4 which I played through several times. Most of the 1st party Nintendo games were excellent on the cube. Too much to choose from!

here's my list...

Most Played:
Mario Kart Double Dash

Best Looking Game:
LOZ: Wind Waker

Best Multiplayer:
Warioware Inc.

Best Single Player
LOZ: Wind Waker

Best Puzzle:
Super Monkey Ball 2

Best Original Game:
Pikmin

Best Racer:
Mario Kart Double Dash

Best Shooter:
Batallion Wars

Best RPG:
FF:Crystal Chronicles

Best Sports
Wave Race Blue Storm

Biggest Dissapointment:
Kirby's Air Ride

Best Game
Warioware Inc.

Worst Game
Wreckless

We'll be compiling a "Best of Gamecube" article this week as we count down the launch for Wii, look out for it on the homepage

Most Played:
Sonic Adventure 2 Battle

Best Looking Game:
Paper Mario: The 1,000 Year Door (I love the graphic style)

Best Multiplayer:
Mario Party (for Co-op!)

Best Single Player
Sonic Adventure DX

Best Puzzle:
Sonic Adventure 2: Battle (I consider the Treasure Hunting levels puzzles)

Best Original Game:
Luigi's Mansion

Best Racer:
Kirby Air Ride!

Best RPG:
Paper Mario: The 1,000 Year Door

Biggest Dissapointment:
Super Mario Sunshine

Best Game
Sonic Adventure 2: Battle

Worst Game
Jimmy Neutron!