Edge momentum shifting
Edge momentum shifting is an advanced technique in Super Smash Bros. Brawl and in Super Smash Bros. 4 prior to update 1.0.8 which involves a character dashing off an edge and using one of their specific special moves to cover a large distance off the edge.
How to perform an edge momemtum shift
Edge momentum shifting can be a challenge to perform consistently, but, once mastered, it can be used either as a vital edge-guarding tool or an additional movement option depending on the character. Not every character can perform the technique and its usefulness will depend on the special with certain specials benefiting more from it than others. To perform the technique, first dash towards the edge, then just as your character reaches the edge, initiate the special move you want to use. The result should be that the character glides a long horizontal distance off the edge. If performed incorrectly, the character will either stop moving if done early, or stall in the air if executed late.
There are multiple known combat applications for edge momentum shifting although its usefulness varies between characters. One of the more common uses is for edge-guarding. Any aerial move can be executed after edge momentum shifting off an edge (assuming you do not land on another platform immediately after), but the most useful of them would likely be to use it off the stage followed by an aerial. Mario for example can perform the technique with his Cape (known as a "Cape Glide") and if his Cape connects, he can follow it up with a forward aerial meteor smash, because, with correct timing, the player can almost always return to the stage using Mario's recovery moves. The technique can also be combined with B-reversing on platforms or stages with no solid walls to give the technique further utility and making the technique useful for quick maneuverability on levels with midair platforms such as Battlefield.
List of characters who can perform the technique
(prior to update 1.0.8)
Video demonstrating Mario's edge momentum shift (Cape Glide)
Video from FireMario149