Directional Influence, or DI for short, is a technique that allows a player to control where they are launched or where they move around in a combo, such as a Smart Bomb or Meta Knight's Mach Tornado. It can help characters live to ridiculously high percentages.
Types of DI
There are three types of DI: Directional Influence, Tap/Smash DI, and Momentum Canceling.
Directional Influence is when you aim for the direction you want to go when being launched. If you are launched vertically, tilt the control stick to the left or right. If launched horizontally, tilt the control stick to the upper left or the upper right, in the opposite direction you are launched.
Tap DI/Smash DI
Tap DI or Smash DI is controlling the direction a character moves in multiple hit moves or combos. This can also be used to escape grabs at high percentages. It also works with electrical attacks and fire attacks. It can be better seen this way in slow motion. To tap DI, tap the control stick in the direction you want your character to move. To smash DI, do the same with the C-stick. If you smash/tap DI enough in a grab such as Yoshi's, you can make the game crash from it.
See the page Momentum Canceling for information.