Stamina Mode

For the stat held by the Pokémon Trainer's Pokémon, see Stamina (stat).

Stamina is a Special Melee option in SSBM, and a Special Brawl option (under the name of Stamina) in SSBB. In both games, it gives all characters a "life bar" of sorts, which must be depleted to defeat characters (rather then KOing them). Once a character reaches 0 HP or are KO'd, they are out of the game.
In Melee, choosing Stamina Mode fixes the stamina at 150 HP; in Brawl, players can select the HP via the Handicap option, ranging from 1 to 300.

Super Projectile Match

Super Projectile Match is an official type of gameplay, originating from Super Smash Bros. Melee and returning in Super Smash Bros. Brawl.

How to perform

This gameplay mode is available for 2-4 players. Choose a Stamina mode battle under Special Melee in VS Mode, and then choose characters that use projectiles. Prohibit all direct attacks, and fight using only projectiles, then fight on a stage that has leveled ground and many places to hide (e.g Hyrule: Temple, etc.)

Notes

This gameplay mode is found on page 49 of the Super Smash Bros. Melee instruction booklet.

Jumps only, arials singled out

Jumps only, arials singled out is an unofficial type of gameplay. You have to be in the air for an amount of time (which finally makes floaty characters have an advantage), and whoever goes to the ground first, loses. When someone touches the ground, one must perform an arial attack on them while they are on the ground. Fox is the worst for this game, but Jigglypuff is the best.