Jumping is a feature introduced in Grand Theft Auto 2 and had been used in all subsequent GTA games except GTA: Chinatown Wars, where instead of jumping, a rolling command is used and presumably functions the same way as a jumping command. In GTA 1, the player could only slide over or under vehicles as opposed to jumping over them, they cannot slide over or under anything else though.
In GTA San Andreas, GTA IV, The Lost and Damned, The Ballad of Gay Tony and GTA V, climbing was introduced so when the jump button is pressed near a low (but not too low), medium sized, reachable or vaultible object, the player will climb onto or over the object. Jumping is usually used to jump over a gap that can't really be gone over if without jumping. In GTA: Chinatown Wars, a rolling command is used, even though it's not actually a jump that's performed, objects can be vaulted over so it acts similar to the jumping command.
In order to jump using the default controls on a PlayStation, "Square" must be pressed. On the Xbox, "X" must be pressed and on PC, the spacebar is pressed (with the exception of Grand Theft Auto: San Andreas where shift is the jump button).
The iOS/Android port of GTA III features a seperate jump button as well, while in the port of GTA Vice City it's mapped to the same button as sprinting, and for the GTA San Andreas iOS/Android version, player will climb/jump over objects automatically if the player makes contact with the object while sprinting.
Jumping and climbing
In some of the first games, the Protagonists couldn't jump and climb over objects such as walls or gates, the protagonist could only jump which was still useful for avoiding collisions with vehicles.
In later games, the protagonists could jump up and climb onto an object. With the same controls as jumping, the protagonist will automatically latch onto the top of an object within reach, climb up it, or if it's a gate or wall, climb over it and land on the other side
In GTA V, while in story mode, if the player jumps then immediately presses "B" on the Xbox or "O" on the Playstation, they can rag doll themselves. This does not work on GTA Online.