In various urban activities, a vault is any type of movement that involves overcoming an obstacle by jumping, leaping, climbing or diving over an obstacle while using their feet, hands or not touching it at all. Although Parkour doesn't involve the idea of set movements, traceurs use similar ways of moving  to quickly and efficiently pass over obstacles.
Safety vault/Step Vault: Similar to a side vault, except that the outside foot is placed on the obstacle and the inside arm is then released, while the outside foot pushes the traceur off the obstacle.
Speed vault: Like a step/safety vault, the outside hand is placed on the obstacle to act as a support,although, the inner leg launches the performer upwards and kicks over the obstacle and the outside foot is forced backwards to force the performer further. The performer lands upon the inner foot and continues running. Or, both legs kick up and over the obstacle like a side vault.
Two-handed/Simple/Side vault: Both hands are placed on the obstacle and the body goes alongside the arms, with the legs parallel to the obstacle.
Gif of a Lazy Vault
Lazy vault: Approach the object from a parallel or diagonal direction. The inside hand goes on the object and you swing the inside then the outside leg over the object in one fluid motion. The other hand is then placed on the object to stabilize and help propel them over the object.
Thief vault: Similar to the lazy vault, except the inside hand touches the obstacle and the outside leg goes over first. This is usually used to change directions fairly quickly.
Monkey vault/Kong vault: Both hands are placed on the obstacle and are used to push the traceur over the obstacle while both legs are pulled up into a squat position with the hips raised to pass over the object. Can also be done in a diving motion over the obstacle.
Double Kong vault: A diving Kong vault with a monkey vault going into the move so that the hands are placed twice on the object.
Dash vault: A jump is performed over an object feet first while the hands are then placed down to carry oneself over. The result will be a sitting position halfway through the vault with the arms used to push off the obstacle.
Kash vault: A combination of the Kong/Monkey and a dash vault. The execution is similar to a monkey vault, except at the end of the vault, the traceur pushes them-self up in order to give some space for the legs to exit as a dash vault.
Speed vault: A jump sideways while one hand is placed down and is used to push oneself forward.
Turn vault: Similar to a side vault but with a 180 degree turn. One hand is usually placed in the underhand position while the other is placed on it normally and a 180 degree turn us performed typically landing in a cat grab.
Lazy turn: A lazy vault, but with a turn to land in a cat position.
Pop vault: The traceur will run at a wall and pop off the wall and propel them-self upward to overcome the obstacle and vault over it.
Gate vault: One hand is placed on top of an obstacle's surface while the other hand is placed on the side of the object and the traceur flips over the object. This move is typically done on gates or walls.
Reverse vault: Performing a backspin over and object while placing either one or both hands on the object to control the motion and to push off.
Reverse kong: A jump similar to a dash vault is performed, but a 180 degree turn is performed before the hands are placed down as in a kong vault and the traceur pushes off the object and turns back around to face forward.
Dive roll: A dive over an obstacle with a roll to withstand the shock at the end.
Drop vault: Any vault followed by a drop.
Wall run: An upward run up a wall. Can also be a run along the wall in a parallel motion to the ground.
Cat grab: Clinging to an obstacle in a squat position.
Cat leap: To jump for distance and go into a cat grab.