What is DI?

DI stands for Directional Influence, a game mechanic present in all the Smash games and applicable to all characters that is not only vital for surviving that Falcon Punch at 70%, but also for escaping all sorts of potential set-ups and combos during the lower percentages leading up to the killing blow. DI makes up a massive part of the metagame across the series, especially in Melee, where a single mistake with your DI can lead to a knee or tipper to the face...or even worse.

The basic explanation of DI is that when an attack lands, you can change the direction in which your character moves slightly (or massively, depending on the attack and the percentage at which it lands) using the control stick (as well as the C-stick). As you might imagine, this can be a life-saver in those situations where you are flying towards the edge of the stage as the result of an attack: where even a bit of extra distance between you and death could save you a stock and quite possibly the match.

Good DI vs. Bad DI

Good DI is getting as close to a 90 degree angle as possible between the direction of DI and the trajectory the attack sends you at without DI. The directional vector of your DI is then added to the directional vector of your default trajectory(but not the magnitude; the magnitude of travel is constant with or without DI, except when hitting against the stage), influencing the resultant direction accordingly. If you would normally be sent flying straight upwards, you would ideally DI straight left or straight right, since this would cause you to go to the upper-left or upper-right of the screen. The general rule is to DI such that you are sent towards the furthest corner of the screen, since the hypotenuse of the right-angled triangle formed between the arrows drawn from you to the furthest sides will always be the largest side by Pythagorean's theorem(it will take more distance to kill you if you travel by the hypotenuse).

Certain moves in Melee, such as Falcon's downwards aerial (referred to hereby as dair), lead to more and more devastating combos the worse your DI is(at some higher percents with some characters, having bad DI when Falcon's dair lands can lead to an unavoidable Falcon Punch to the face). In cases like this, good DI becomes a matter of prediction: the goal is to DI such that your character will go in a direction that either your opponent didn't anticipate or that puts you outside the range of their best attacks.

As you get to higher percentages and you take powerful hits just above the stage or on it, it can be useful to DI towards the closest part of the stage. By doing so, you will land against the stage, enabling you to tech (as described in Raven Dreamer's answer) to reduce or completely cancel momentum and hopefully survive.

This is not to say that DI is only applicable to combo setups or powerful killing blows. It applies to every single attack in the game. Let's say that Fox has you pinned against a wall with his multi-hit jab(rapid kicking). You could simply choose not to DI, and take a completely unnecessary extra amount of damage, or you could DI up and escape much more quickly. Learning about different types of DI will show you how moving the control stick in quarter-circles around the general area you wish to DI will help you escape attacks such as these even sooner.

Learning to perform DI

DI is trained as an instinct within competitive players. Even with all the theory and understanding of how DI works, it's still unlikely that you'll immediately possess the reaction speed to anticipate the hit before it lands and DI well. However, by remembering about DI and by applying it slowly and gradually to your playstyle(i.e. letting yourself take obvious hits at higher percentages to see what trajectories are good and what trajectories are bad), you will develop the instinct to hold on to life at the brink of defeat, so don't rush or try to absorb too much information about DI at once. Master the basics before moving on to the more advanced details, which will be included in a later update of this answer.

In short, DI OR DIE.

Tripping During Movement

Tripping during movement is (very unfortunately) COMPLETELY RANDOM. It is, however, caused by your character moving forward on the ground.

There are a few common assumptions made about tripping that are wrong. Facts about tripping established through stats:

• The person in the lead will not trip more.
• Tripping will not occur more often when you turn/skid/start a dash.
• Pressing the joystick perfectly forward will not cause your character to not trip.
• Being at a higher percentage/being hit by more moves will not cause your character to trip more often.
• Faster/different characters will not trip more often than other characters.

Tripping By Attack

Some moves, or some hitboxes on some moves, will cause tripping in the range of a lower damage limit to an upper damage limit determined by the weight of your character. Smashboards regular Mugwhump has made a very amusing guide detailing the specifics of what moves and what percentages will cause characters to be tripped by an attack. The basics are that when you are hit by a move which can trip in that damage range, you enter into a HUARGH state: your character falls down and is forced to either get up or roll. Beyond that range, the character will enter into a WHARRGARBL state: they will leave the ground completely from the impact.