The wall jump is performed by jumping towards a wall and then pressing the control stick in the direction opposite the wall. It must be performed before the character reaches the peak of his jump. A wall jumping character experiences a short period of invincibility at the moment he kicks off the wall. One cannot wall jump forever though, as the height one gains from each successive wall jump decreases until they actually lose height.
Only a few characters can wall jump.
In Melee, these characters are: Captain Falcon, Falco, Fox, Mario, Pichu, Samus, Sheik, and Young Link.
In Brawl, these characters are confirmed to have the Wall Jump ability: Diddy Kong, Donkey Kong, Lucario, Fox, Mario, Pit, Samus, Sheik, Zero Suit Samus and Sonic.
In Brawl, a few characters are capable of sticking to a wall, allowing them more time to do their wall jumps.
Everybody, however, can perform a wall tech jump, which garners the same results as a wall jump, but must be performed after wall teching.
Mario, Samus and Sonic have performed wall jumps in their own series; Mario in Super Mario 64, Samus in Super Metroid and Sonic in Sonic heroes.