In various urban activities, a vault is any type of movement that involves jumping or leaping over an obstacle while putting weight on it with one's hand(s). Although Parkour doesn't involve the idea of set movements, practitioners (traceurs) sometimes use similar ways of moving  to quickly and efficiently pass over obstacles.
People have created many names for variations on the theme:
Lazy vault: One hand is used to swing both legs over an obstacle (the inside hand touches the obstacle and the inside leg goes over first). The other hand is placed behind your body to create stability and a way to further propel your body.
Kong vault: A monkey vault, except that you dive perpendicular to the object - only placing your hands down at the end of the obstacle. The hands are then used to push down and the legs are put in a monkey position. Also known as the King Kong or the dive Kong.
Dash vault: Jump first, then place hands 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 and dash vault. The execution is similar to a Kong vault, except at the end of the vault, the traceur pushes himself/herself up in order to give some space for the legs to exit as a dash vault.
Two-handed vault/Side vault: Both hands are placed on the obstacle and the body goes alongside the arms, with the legs parallel to the obstacle.
Safety vault/Step Vault: Similar to the two-handed vault, except easier. Place both hands on the obstacle, then 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: A jump sideways but in the center of the obstacle, while one hand is used to push oneself forward. This is similar to the lazy vault; however the inside hand does not touch the obstacle.
Thief vault: Same as the speed and lazy vaults, except the inside hand touches the obstacle and the outside leg goes over first. This is usually used to reverse directions over a rail/wall fairly quickly.
Double Kong vault: Similar to the kong, but the traceur pushes himself/herself off the obstacle twice before finishing the vault. This is used for especially long obstacles.
Turn vault: Place one hand on the obstacle, preferably in the underhand position, place the other on it normally, then turn over 180 degrees.
Pop vault: Run towards an obstacle and place one foot on it, kick upwards, grab the top of the obstacle and go over it.
Barrel vault: A straight legged side flip over the obstacle with one hand placed on it for balance and control.
Reverse kong: Jumping over the obstacle while turning around, the hands are then used to push off the obstacle to gain more distance. Technique can be thought of like a reverse dash vault and is specifically used in parkour to remove momentum.
Many activities have built on more common movements and given names to many other types that can be considered vaults.
Reverse vault (not to be confused with the 360 side vault): Place one or two hands on the obstacle and perform a backwards spin over it.
360 vault: Like a speed vault, but with a 360-degree turn about the hand.
Cast bomb: Backwards kong to back flip.
Cast vault: Backwards kong.
Kong gainer: A kong vault, but with a back flip in the air.
Dash bomb: A dash vault, but with a front flip after the dash.
360 degree underbar vault.
Double kong vault: A kong with two taps; useful to get over two obstacles at once.
Triple kong: A kong with three taps.
Gate vault: One's weak hand is placed on top of an obstacle's surface while the strong hand goes below for strength in rotation over at a 45-degree angle.
Sky vault: A one-handed cartwheel over an obstacle.
Vert vault: Hands are placed in a regular vault fashion but one leg is kicked up over one's head.
Rocket vault: Done in a lazy fashion, both legs go together and are perfectly straight, and the feet are grabbed by the traceur's hand.
Rail flip vault: Two hands are placed on top of the obstacle and a front flip / forward handspring is executed.
Shoulder vault: To roll over an obstacle as efficiently as possible. Usually done after spoiling a move.
Wall spin: Run at a vertical surface at an angle and spin oneself 360 degrees around the surface.
Reverse vault: A kong vault, but when the hands are placed, a twist is done to go further over the obstacle.
Lazy "turn”: A lazy vault, but with a turn to land in a cat position.
Dive roll: A dive over an obstacle with a roll to withstand the shock at the end.
Carriage: A wall walk, but with a Kong right at the peak of the walk.
Palm spin: A thief vault, but hands spin the body in a 360-degree motion.
Barrel: A double-leg reverse over an obstacle with one hand.
Star: A cartwheel over an obstacle.
Thief 360: A thief vault but with a 360-degree twist. It is usually immediately followed by a roll.
Inside monkey: A monkey vault but the legs are outside the arms; the legs are still bent monkey-style.
Inside dash: A dash vault but the legs are outside the arms.
Underbar 180: Underbar, but hands are used to twist body in a 180-degree motion.
Double jump: Practitioner uses a higher object/obstacle to jump from to gain more height/distance.
Dash 360: A dash with a 360-degree twist.
Dash bomb barani: A dash bomb with an aerial, or barani, instead of a front flip.
Weave: Practitioner goes through an opening (for example, between two bars) and quickly comes out the other side.
Blisto: Like a regular vault but with a 360-degree twist. A Blisto with a 720 is a "blast".
Wall climb: Like an upward wall run, but hands are used to push oneself upward.
Cat grab: Clinging to an obstacle in a squat position.
Cat leap: To jump for distance and go into a cat grab.
Pop vault: One of the most basic and popular moves. Traceur will run at a wall or fence and "pop" (kick) and propel himself upward to overcome the obstacle, which is usually taller than them. (Ex. Pop regular = kick up wall and then do a regular.)
Monkey roll: A monkey vault followed by a roll, without landing on your feet before the roll.
Butterfly vault: Similar to the reverse vault, but legs swing in either a clockwise or anticlockwise direction. It is called the butterfly vault due to its twist Relation with the Butterfly twist.
Obstacle vault: This movement just slightly happens to be a vault it's very similar to a dash vault but only you don't put your hands on the wall or rail and sort of squat forward over the object in a forward motion.
Split vault: Performing a split in a mid air vault.
Kick vault: Kicking one leg out over the wall then rest both hands on the wall and push off wall getting other leg over.
Leopard vault: Leaping from a higher object to a lower one placing slapping hands on lower wall and pushing off.
Descent vault: 180 vaulting over an object e.g. wall and dropping down.
Setter vault: Performing a 180 onto a wall which gives you this "set up" to perform a backflip for example to get off the wall.
Air bomb: similar to the dash bomb but only rolling sideways of a wall nice and tucked and then leads into a propelling sideways flip descent.
Slide vault: If you come across a large wide surface to long enough to vault then a slide vault is when you hop onto the object and slide off it tapping your hands on it as you leave the obstacle.
Spring forward: A rail flip but with a spring flip and legs are spread out.
Scissors: In a lazy motion, but legs move side by side in a scissor like cutting motion.
Superman: In a kong motion, but hands do not touch the obstacle.
Cannon: A jump but legs are in monkey form and arms go around cupped legs.
Underbar 540: Underbar but with a quick 540 twist.
Gate flare: Like a sideways gate vault motion, with legs perfectly straight.