DI is simply using your control stick to control where you fly/fall. If someone says "DI right", what it means is simply "hold your analog stick right".
You're actually supposed to DI perpendicular to the direction you are flying in. So if you're sent upward, DI to the side, and if you're sent sideways, DI up or sometimes down. The idea is that you aim yourself toward one of the corners of the screen, because these are the farthest blastzones. You can't actually reduce the the distance you fly when you're hit, but you can influence the direction. Hence, Directional Influence. DI-ing in the opposite direction in which you are flying does nothing.
SSK is right about the aerial, however. When you're sent flying, you want to use your fastest aerial (fair or dair) once while DIing in the appropriate direction. This is done because after the aerial is complete, you regain control of your character and you can influence your direction even further by jumping or fastfalling. Practice these things and you will live long enough to blaze their pants off with your Aura Spheres the size of Donkey Kong.
There's also Smash DI, which I might explain later if I feel like it. There are a lot of videos and threads made about this subject. You might also want to check those out if you still don't understand. I know it can be a hard subject to grasp.