Directional influence

How to DI
Directional Influence is the control the target of an attack has over the attack's trajectory. You DI by pressing the control stick in any direction when being hit by an attack. Different directions yield different results.

Examples
If a 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 you are hit with any move, there are a few frames in which you can input control stick directions which have a much greater effect on the trajectory of the attack than normal DI. It is possible to do things such as survive a Marth f-smash at greater than 100% 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 involved pointing both the joystick and the C-stick in opposite direction so the player can DI in different directions. This was later found out to be false, and what players were actually doing was Smash DIing.

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.

Links
Doraki's quide to DI, Teching, and Crouch Cancelling