The air jump (airjump) is a jump done off the air while not in contact with a platform. Sometimes known as the double jump for it's similarity to the mid-air jumps often found in platform games, the air jump allows a mouse to reset their vertical acceleration to give them better control midair, and jump distances otherwise impossible to reach.
Mice reset their ability to jump after landing on a platform and do not lose it until the jump is used, regardless of whether the mouse is currently standing on a platform or not. This means that for every time you "fall" on a platform you will receive the ability to jump once, anywhere. This is also how mice are able to wall jump.
Air jumping is commonly done to extend a jump's horizontal distance, sometimes called longjumping or fakefalling. This is done by running off a platform and jumping to safety from midair. It is often done in conjunction with walljumping.
By running off the wood and falling, a mouse will have already gone a certain distance horizontally to their destination. Then, by jumping in midair, the mouse then is able to complete the other end of their jump. A normal jump would mean that the mouse simply jumps and falls.
The exact mechanics and perfect jumping "sweet spot" is rather unknown. However, airjumping will break the mouse out of its fall, regardless of how fast it is falling, making it most efficient to jump when the mouse has lost all upward momentum and has reached terminal velocity. It is usually best to jump when halfway to your destination.
Another way in combination with the above method for jumping far is to stand on the very edge of the block you jump from. Now, run to your destination and do the above method. You will find that you can jump ever FURTHER than just ordinary airjumping. This method is often used in bootcamp.
 Turning in Midair
In some instances, mice must fall along a wall and then jump under the wall to land safely. To do so, mice fall until when below the level of the wall, then jump towards it.
Airjumping is useful when combating shaman attacks or other flying objects. Mice who are able to airjump can fall to dodge items such as a drag ball shot, then immediately jump and walljump back up. Even when hit, an airjumping mouse can fly diagonally into the air, then airjump back to safety.
 Repulsion Wood (Lava)repulsion wood. This is best done by holding the jump button before touching the red wood, and continuing to hold it along with the directional key until safely above non-repulsive wood.
Contacting the surface throws mice upward, but also resets the jump, allowing another jump before flying too high. Airjumping returns a mouse's upward velocity to that of a jump, much slower than that of launching up from the repulsion wood, allowing gravity to drop the mouse back to down. Holding both a directional key and the jump key utilizes the game's "autojump" feature, which allows mice to jump immediately after touching down.