Directional influence
Directional Influence (usually abbreviated DI) is the control the target of an attack has over the attack's trajectory. The player DI's by pressing the control stick in any direction when being hit by an attack. Different directions yield different results.
Examples
If Fox up-smashes you, you can tilt the control stick left or right to travel mainly to either of the directions. If you DI to the right when Up-smashed, your character will travel more to the right than usual, making the probability of your survival much higher.
DI is essential for escaping many combos, especially chain grabs. A good DIer can also survive at higher percentages.
Smash DI
Immediately after the player is hit with any move, there are a few frames in which he can input control stick directions which have a much greater effect on the trajectory of the attack than normal DI. It is possible to, for instance, survive Marth's forward smash at greater than 100% damage using this tactic.
A subtactic is the quarter-circle DI, which involves crouch-cancelling an attack followed by smash DI towards the stage, 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.
Another subtactic 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 sexy, 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.
It is also possible to perform Multiple Smash DIs, which results in the character to move in several consecutive directions, each having their own frame of movement. 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.
