Vaulting is a fancy parkour was of saying "jumping over an object or obstacle."
The biggest difference is, that vaulting is much more efficient than just jumping over an obstacle.
The goal of a vault is to pass over an obstacle and lose as little momentum as possible to keep maximum velocity.
One of the most common vaults, is called the Kong.
There are three different techniques to start a Kong, that each lead into a more advanced variations of a Kong.

The Kong in general, starts with a run up, an arm swing, a forward lean jump, precise hand placement, feet returning under body, and the landing.
Run up
The run up is what it sounds like, running up to the obstacle. The important things to take into consideration are speed, and distance to be traversed.
This determines what kind of Kong you will be performing, by placing your body in the right position to keep your center of mass where you want it,
as well as giving you a velocity large enough to pass over the obstacle.
Arm Swing
The arm swing happens as soon as you push off to jump. You swing your arms forwards, and this helps propel the body as well as move the center of mass
Forward Lean Jump
Leaning forward, making sure your feet are split, and jumping along with the arm swing are all important.
Leaning adds the necessary rotation of the body.
Splitting your feet is important because it effects distance and rotation significantly.
Jumping along with your arm swing adds to your momentum, adding to the efficiency of the Kong.
Precise Hand Placement
Placing your hands on the object is important because this determines the distance, as well as the efficiency of your Kong
Arm placement also counter-rotates your body back to an upright position
Feet Return
After projecting yourself off of the obstacle with your arms, your body will rotate back towards an upright position, allowing you to move your feet back under your body.
Landing
How you land, determines how much velocity you retain after your Kong.

These videos explains in VERY great detail, the Kong.
https://www.youtube.com/user/ParkourScience/videos