Directional influence (usually abbreviated DI, which can be used as a verb) is the control the target of an attack has over its trajectory. Each attack sends its target in a particular direction, depending on the attack itself and on the target's weight and falling speed; DI can be used to alter, but not completely change, this trajectory. This change, however, can be vital to surviving high-power attacks such as Fox's up smash, and for escaping combos such as Jigglypuff's Space Animal Slayer.
The player can DI by pressing the control stick in any direction during or just after being hit by an attack. There are three types of directional influence: regular DI, smash DI, and automatic smash DI. The type of DI changes depending on when the player presses the control stick.
Normal DI
DI is essential for escaping many combos, especially chain grabs. A good DI can also survive at higher percentages.
Smash DI
Smash DI is performed during the frames of hitlag in which the target is immobile after being hit by the attack. The length of the hitlag varies from attack to attack, from 2 frames to 18 frames (Samus's Charge Shot). Smash DI allows the character to move, which can slightly affect knockback, and can move the character into a wall or ceiling to allow wall and ceiling techs. By pressing different directions on the control stick, the player can Smash DI multiple times. This is best represented in the Perfect Control video, at approximately 2:36, and with a frame-by-frame version at about 3:51, during the end credits.
A subtactic is the quarter-circle DI, which involves inputting multiple Smash DIs by rotating the control stick 90 degrees, resulting in a drastic decrease in knockback. Quarter-circle DI was introduced to America by Captain Jack and was known for a while as Japanese DI.
Smash DI is the only type of DI that exists in the original Super Smash Bros.
Automatic Smash DI
Automatic Smash DI functions similarly to Smash DI, but it occurs the frame after the attack's hitlag ends. The character will move, but not as far as in normal Smash DI.
A tactic is double stick DI which involves pointing both the control stick and the C-stick in 2 different directions so the player can alter the trajectory with the control stick and use the C-stick to control the Automatic Smash DI into a different direction from the direction being held on the control stick. This was later found out to be quite useful, especially when used to get the lowest angle of knockback while ASDI-ing straight downward to tech along the ground and survive high damage hits well beyond what is possible by taking the hit into the air.
Links
Doraki's quide to DI, Teching, and Crouch Cancelling A video that demonstrates the usefulness of crouch canceling and DI