Jumping is a feature introduced in Grand Theft Auto 2 and had been used in all the other GTA games furtherwards except GTA: Chinatown Wars (instead of jumping, a rolling command is used and presumably the jump command).
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 vaultable 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)
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.
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.