From the Super Mario Wiki
The Wall Jump (also known as the Wall Kick) is a move that several characters can use in various games. Within narrow passageways, characters can use the move to gradually ascend by jumping from wall to wall, gaining a little more height with each jump. Thus, the move can be used to attain great heights, cross wide gaps, reach inaccessible platforms, or prevent falling into a bottomless pit. In most games, the move is performed by pressing the jump button while touching or sliding down a wall. It has been used in many platformer games.
The move first appears in Super Mario Bros. as a glitch. If Mario lands on a wall on a 16×16 boundary (so that his feet is on the top-corner of a block he's standing on), the player is able to cause him to jump from the wall if they have frame-accurate timing. A Wall Jump glitch also exists in Super Mario Bros. 3, Super Mario World and Yoshi's Island.
Glitches excluded, the move first appears in Super Mario 64, and appears later in its remake, Super Mario 64 DS. There is a level in Cool, Cool Mountain called "Wall Kicks Will Work" which derives its name from the move. Mario is the only character who can Wall Jump in these games.
The move also appears in Super Smash Bros. Melee, Super Mario Sunshine, New Super Mario Bros., Super Mario Galaxy, Super Smash Bros. Brawl, New Super Mario Bros. Wii, Super Mario Galaxy 2, Super Mario 3D Land, New Super Mario Bros. 2, New Super Mario Bros. U, New Super Luigi U, Super Mario 3D World, and Super Smash Bros. for Nintendo 3DS / Wii U.
In Super Smash Bros. Melee, Super Smash Bros. Brawl, and Super Smash Bros. for Nintendo 3DS / Wii U, several characters can Wall jump. Since Super Smash Bros. Brawl, some of these characters are able to Wall cling as well.
In Super Mario Sunshine, Mario is able to jump off walls at different angles depending on the angle from which he jumps at a wall. This allows him to jump off a wall at a perpendicular wall, a technique which is necessary for some levels in the game, particularly "The Shell's Secret".
In Mario Superstar Baseball, this move is used as a special fielding ability. Mario, Luigi, Bowser Jr., Baby Mario, Baby Luigi, and Toadette can use the move in this game. When a character jumps near a wall, they take a great leap off of it, covering distance at a vertical angle. The move is used to catch low home run balls. In Mario Super Sluggers, this special ability is absent, possibly replaced by the Buddy Jump move.
In Super Mario Galaxy and Super Mario Galaxy 2, Mario/Luigi's Wall Jump has more of a stick to it, meaning he can stay attached to walls a tad longer than in previous games. He can also perform a Spin Jump while Wall Jumping. In Super Mario Galaxy, there is a secret level in the Beach Bowl Galaxy called "Wall Jumping Up Waterfalls", where Mario has to transform into Ice Mario and Wall Jump upward using parallel waterfalls as walls in order to reach the Power Star. In Super Mario Galaxy 2, the Mailtoad mentions that the Toad Brigade Captain has the ability to Wall Jump as he was able to scale a canyon in pursuit of The Chimp in the Fluffy Bluff Galaxy.
In New Super Mario Bros. Wii and New Super Mario Bros. U, Mario, Luigi, Yellow Toad, and Blue Toad are capable of Wall Jumping. In New Super Luigi U, Luigi, Yellow Toad, and Blue Toad can perform the move (Mario does not appear in the game). In New Super Mario Bros. U and New Super Luigi U, the player can perform a Spin Jump while Wall Jumping.
In Super Mario 3D World, Mario, Luigi, Toad, Princess Peach, and Rosalina are able to Wall Jump. In this game, if the player performs a single Wall Jump (or half of a single Wall Jump) while wearing the Cat Suit or Lucky Cat Suit, they can remain hanging on the wall for a short period of time after jumping. However, if the player has already hung onto either or both of the two parallel walls for as long as the Cat Suit or Lucky Cat Suit will allow, or if they Wall Jump again after executing the first Wall Jump, they will begin sliding down the wall as normal (though they can still Wall Jump upward).
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