Smash DI :
When you take a hit, there are frames of hitlags where you're frozen and stay in place before being sent flying. The number of hitlag frames varies between 2-3 up to around 18 with Samus's Fully Charged Shot, or burner hits in BtT stages.
During those frame, you can move around using Smash DI.
If you smash the Control Stick in one direction, your character will move instantly in that direction. It doesn't go very far, but it can still save your life sometimes.
If the hit is techable, Smash DI can be used to move yourself into a wall or a ceiling and you'll be able to tech. If you miss the tech, you might bounce on the wall/ceiling
However Smash DIing into the ground can't make you tech (it would clearly be broken)
Forbidden Smash DI :
If you are on the ground and are hit by a non-techable move that sends you horizontally or downwards, you won't be able to Smash DI up.
I call this Forbidden Smash DI. It would in fact be broken because you would take the hit airborne, you would land on the ground without any stun.
The developpers saw this and prevented that from happening.
Forbidden Smash DI makes sure you stay stunned on the ground when you're hit by Fox's Aerial Down A, Fox's shine or any other relatively weak hit nailing you into the ground.
(However I've seen CPUs breaking that rule in Adventure mode )
Multiple Smash DIing :
If you are fast with the control stick you can sometimes input several Smash DIs
This is basically what you do when you use the so-called "Quarter-circle DI".
In the Perfect Control Video, there is a clip where a Mario on Jungle Japes takes a Charge Shot from the left, use Smash DI on each frame of hitlag alternating between down-left and up-left, and finally Smash DIs up into a ceiling and wall-tech-jumps there.
Regular DI :
After the hitlag, your character starts flying from the hit.
Regular DI will influence your trajectory, it has been known for more than a year now.
DIing along the same line of the defaut trajectory doesn't affect it at all. What does work and what saves your life at high % is having a DI perpendicular to the defaut trajectory.
Automatic Smash DI :
An Automatic Smash DI takes place there on the 1st frame after the hitlag.
It's automatic because you don't have to smash anything to do it, the game just reads the position of your control stick or C-stick on the last frame of hitlag.
The C-stick outprioritizes the control stick. You can hold 2 different directions, the game will use the C-stick for the ASDI : you can DI one way and ASDI another way.
The ASDI is exactly like Smash DI except that it goes less far, and you don't see it very well because you're sent flying at the same time.
On techable hits, it can make you tech on walls or ceiling just as well.
One very important thing is that you can tech on the floor as well if the ASDI makes you go down on that frame.
The instant ground techs on other moves than Fox's shine all come from this.
ASDI Down, the true Crouch Cancelling effect :
If you're grounded and take a move that sends you upwards but not very fast, ASDIing down'll make you go downwards on that frame. You'll then either bounce/tech if the hit is techable, or just land with no stun.
If you're grounded and take a move that sends you down, if it's untechable, Forbidden Smash DI applies and you'll stay stunned on the ground whatever you try to do (DI doesn't seem to work in those cases, too.)
If it's techable, you'll see a green flash on the ground at the beginning of the hitlag and you'll be sent up. You can try to ASDI down that and you may bounce/tech that hit.
For example, Fox's Aerial Down A sends you downwards and is never techable.
Falco's Aerial Down A sends you downwards and becomes techable at low/middle %
Falco's Phantasm sends you up if you're grounded but sends you down when you're in the air, and it's always techable.
Fox's shine sends you horizontally when you're grounded.