Directional Influence, or DI, as it is more often called, refers to the ability to control the aerial momentum of your character. Simply hold the analog stick in a direction to set your DI to that direction. This is very useful when it comes to recovery, because with proper DI, it is possible to survive blows you would otherwise not have survived. In general, you want to DI perpendicular to near fatal blows: Should you be potentially sent flying left off the screen, you would want to DI away from both stage boundaries. Unlike Melee, the stage boundaries are shaped like an octagon. For this reason, you want to change your trajectory course towards the corners of the upper parts of the octagon. Using DI downwards from fatal blows often puts you below the stage line and makes it more difficult to recover. Generally, you're going to want to DI upwards when hit by fatal blows, although for vertical launches it will be better to apply DI towards the left or right.
Smash DI is another form of DI that is best used for multi-hit moves. Smash DI is performed by smashing the analog stick many times in the direction you wish to DI. This can help you escape from multi-hit moves such as infinite jab combos. On a GCN controller, the C-stick is capable of applying Smash DI (Though not standard DI), so it is best to use both sticks in tandem. Smash DI has more effect than standard DI, and for best results, Smash DI should be applied and then held to lead into standard DI on fatal blows. However, Smash DI must be applied as the blow lands.