Super Mario World (SNES):

Mariothon: the basic aim is to get as far in the game as you can in fifteen minutes. You'll be ranked based on which level you get to. In the event of a tie, players will be ranked on their Score.

Super Mario Kart (SNES):

Race: Players will be split into two player head to head races in a tournament style competition. There will be a match between the two losing semi-finalists to see who gets the bronze medal.
Battle: Similar to the above, except in Battle Mode instead of Race Mode.

Donkey Kong Country (SNES):

Mariothon: the same as the mariothon for Super Mario World, except ties will be divided based on how many times they collected certain symbols, like the K O N G letters.

Super Street Fighter II Turbo (SNES):

Tournament: Players will fight each other in a first to three style match, with the winner progressing to the next round. There will be a match between the two losing semi-finalists to see who gets the bronze medal.

Super Mario 64 (N64):

At the moment we only have one event for this, but I might add some more.

Sprint: On Bob-omb Battlefield, each player has three attempts to get from the beginning of the level to the top of the mountain as quickly as possible, by whatever means they can, excluding cannons and the Wing Cap. Time begins as soon as the "starting pistol" goes off and ends the moment Mario touches the top of the mountain.

Mario Kart 64 (N64):

Race: same as Super Mario Kart, except with four players to a race.
Battle: same as Super Mario Kart, except with four players to a battle.

Pokemon Stadium (N64):

Normal Battle: players each choose six pokemon from the Rental list, and then play each other in a full battle, tournament mode. The two losing semi-finalists play each other for a bronze medal.
Little Cup: similar to Normal Battle, but with restrictions on which pokemon can be chosen. List upon request.
Baker's Decathlon: this game has an excellent minigame mode. Players will each participate in the nine minigames, and receive points based on their performance. The top three scores at the end receive the medals.

Goldeneye (N64):

Deathmatch: players are placed four at a time into a deathmatch with the default multiplayer settings. They progress by winning their match to the next round. In the final, the players will the best three scores take the medals.

Diddy Kong Racing (N64):

Race: same rules as Mario Kart 64.

Super Smash Bros. Melee (Gamecube):

Normal Melee: Players fight each other one vs one in a tournament style competition, with the winner progressing to the next round in each match. The matches are a first to three wins format. All items are turned on.
Super Melee: Same as above, except with four players to a battle rather than two. First to three still applies.

Mario Golf: Toadstool Tour (GC):

Tournament: Players play the first course and the top four scores, including ties, qualify for the second round. The second round is played on the second course and the player with the best aggregate score from the two rounds wins. If there are still ties, those players play at the third course until one player is ahead.

Mario Kart: Double Dash (GC):

Race: same as Mario Kart 64.
Battle: same as Mario Kart 64.

Mario Power Tennis (GC):

Singles: Players play each other in a three set match, in a tournament style competition, with the victor progressing to the next round. The final is a five set match.
Doubles: Players find a partner to enter alongside. Other than that, the tournament follows the same format as Singles.

Mario Smash Football (GC):

Tournament: players are placed into groups of four and proceed to play each other once each. Three points for a win, one for a draw and none for a loss. The top two teams in each group proceed to a knock out competition.

Super Monkey Ball (GC):

Monkey Fight: players are placed into four person heats, with the best player from each heat proceeding to a final bout. Each match is first to three, with second and third position in the final being determined by points.
Monkey Target: our equivalent of archery. Players aim to score as many points as possible by launching a monkey at a target. The best three scores from a ten round match with items turned off get the medals. In the event of ties, a play off shot is taken.

Wario Ware: Smooth Moves (GC):

TBC

Other games that might take place if people are interested include:

Soul Calibur 2
Dr. Mario
Super Bomberman 2
Worms Armageddon
F-Zero GX
Star Wars: Episode One Pod Racer