From the Super Mario Wiki
Jumping is Mario and Luigi's trademark move. While it was originally only used to evade obstacles in the original Donkey Kong game, it has been Mario's primary method of attacking since Super Mario Bros. Depending on the type of game, Mario and other characters jump for a variety of different purposes.
 Donkey Kong
In the original arcade version of Donkey Kong, Mario had to jump over the barrels that Donkey Kong threw at him. He could only jump high enough to leap right over the barrel, in contrast to the impossibly high jumps in the Super Mario Bros. series. If Mario landed on a barrel in the game, he would die. In the early concept of Donkey Kong, Mario originally wasn't able to jump. It was implemented when Shigeru Miyamoto and his team thought "If you had a barrel rolling towards you, what would you do?".
 Donkey Kong (Game Boy)
In Donkey Kong on the Game Boy, Mario has a greater variety of moves related to jumping. If the player jumped, then pressed down on the d-pad, Mario would make a handstand. From the handstand position, Mario could perform high jumps and wide jumps depending on the situation. Unlike all other 2D Mario titles, Donkey Kong would also punish the player for falling too far, resulting in Mario losing a life. This jump system has also been used in Mario vs. Donkey Kong.
 Mario Bros.
In Mario Bros., jumping was the primary attack of Mario and Luigi and the only way to defeat enemies. By jumping at the platforms on which the enemies were standing on from below, the player could turn the enemies around. The enemies would then have to be finished off with a kick. Unlike future games, stomping on enemies is still deadly in Mario Bros. Because of this, the Shellcreepers have been changed to Spinies in all remakes of the game to remind players not to stomp on them.
Since Super Mario Bros., jumping has been a much more integral part of the Mario series, and is no longer used solely to jump over gaps and enemies. In later games of the series, Mario and other playable characters use a greater variety of jump moves which can be used to reach certain areas or get bonus items from special blocks. These include the Spin Jump from Super Mario World, the Ground Pound first seen in Super Mario World 2: Yoshi's Island, and even the Double Jump, Triple Jump, and Wall Jump introduced in Super Mario 64.
 Super Mario Bros.
Super Mario Bros. features various types of Brick Blocks which Mario can find coins in, as well as ? Blocks that contain Super Mushrooms and Fire Flowers. To break these blocks, the player must jump while under them. Mario can also defeat enemies like Goombas and Hammer Bros. by stomping on them. Koopa Troopas are not defeated by the stomp, but hide in their shells, which can then be kicked by the player.
 Super Mario Bros. 2
In Super Mario Bros. 2, jumps served a different purpose than in all other games, due to the game using the only slightly altered Yume Kōjō: Doki Doki Panic as its engine. The characters in the game could not defeat any enemies by stomping on them. Instead, they could stand on them and pick them up by using the B button. Players could even land on Birdo's eggs and throw them at Birdo. Enemies and items could be picked up and thrown to knock out other enemies. This system has been carried over to Donkey Kong on the Game Boy.
 Super Mario Sunshine
In Super Mario Sunshine, Mario's jump can defeat most enemies. When using F.L.U.D.D.'s Hover Nozzle, Mario can float higher and longer. In addition, if Mario loses health, he sounds weaker when he jumps. Also, if Mario jumps over a gap or a void, Mario's jumping sound sounds different.
 Mario Kart series
In Super Mario Kart, players could hop with the press of a button. The Feather allowed them to do a high jump. Jumps can be used to cross gaps and to take shortcuts. In later games of the Mario Kart series, jumping is usually only possible when using ramps. In Mario Kart Wii and Mario Kart 7, it was possible to do tricks after jumping off, giving players a small boost.
 Super Mario RPG: Legend of the Seven Stars
In Super Mario RPG: Legend of the Seven Stars, the Jump was a special move that requires Flower Points. It does additional damage as a Timed Hit, and its power increases with the Jump Shoes. Mario can use this move from the beginning of the game on. Mario's jump was also used by many characters in the game to verify his identity. Everyone also seems to think of it as his most defining characteristic.
 Paper Mario series
Jumping is present in every installment of the Paper Mario series.
In Paper Mario and Paper Mario: The Thousand-Year Door, jumping is Mario's primary attack he can use from the beginning of the game. It can be used to defeat any kind of generic enemy like the Goombas, but some enemies like the Spiked Goombas would prevent jump attacks with spikes, and the player would have to use the Hammer attacks instead. Jumping is also used on the overworld for platforming elements. In both games, Mario can increase his jump power and abilities with the Super Boots and the Ultra Boots.
In Super Paper Mario, all four characters have the ability to jump on foes as their primary means of attack.
Paper Mario: Sticker Star features multiple Jump stickers that can be used in battle, each with various strengths. There are seven different types of Jumps, each with their own variations: the Jump, Iron Jump, Hopslipper, Infinijump, Line Jump, and Clone Jump.
 Mario & Luigi: Superstar Saga
Mario & Luigi: Superstar Saga used the same concept for jumps as the Paper Mario series. On the overworld, Mario and Luigi could also perform the High Jump and the Spin Jump to reach higher areas. Many Bros. Attacks would also use jump moves.
 Mario & Luigi: Partners in Time
In Mario & Luigi: Partners in Time, the jump command was identical to how it was in Mario & Luigi: Superstar Saga, however, the piggyback jumps with Baby Mario and Baby Luigi increased the number of jump commands.
 Mario & Luigi: Bowser's Inside Story
In Mario & Luigi: Bowser's Inside Story, the jump works similar to how it did in Mario & Luigi: Partners in Time, however Mario and Luigi perform both jumps instead of their baby versions performing the first and the adult Bros. the second. During battle, Mario and Luigi can also jump on most enemies.
 Mario & Luigi: Dream Team
In Mario & Luigi: Dream Team, the Jump attacks for Mario and Luigi are identical in game-play to the ones in Mario & Luigi: Bowser's Inside Story. In the over-world, the game also featured the Spin Jump and the Ball Hop, a move very similar in style to the High Jump from Mario and Luigi: Superstar Saga.
The Jump attack is also used during the Giant Battles by an enlarged Dreamy Luigi. This involves Luigi first bobbing down as if he was about to spring into the air, where the player must wait until Luigi flashes to slide up quickly on the Touch Screen. Perfect timing will allow Luigi to land firmly on the target and perform a second jump on them if the player succeeds in repeating the command. Perfect timing with both jumps will yield in an Excellent! rating, perfect timing on only one will yield in a Great! rating, sliding up at the wrong time will result in a Good! rating and not sliding up at all will result in an OK! rating.
Mario - Press the A button as you land to jump again. Time two jumps for an Excellent!
Luigi - Press the B button as you land to jump again. Time two jumps for an Excellent!
Giant Battle - Charge up for extra oomph!
 Luigi's Mansion: Dark Moon
In Luigi's Mansion: Dark Moon, Luigi can jump while he is sucking up a ghost to avoid attacks from other ghosts. This is executed by pressing the button while using the Poltergust 5000 to suck up the ghost.
 Super Smash Bros. series
In the Super Smash Bros. series, a jump can be performed by all characters. Each character has a different jump height and different window of time where they crouch before they jump.
 List of Jumps
 Names in Other Languages