I have always wondered this. Why is the damage done in percents, and what does 100% damage correspond to?
In most ways, 100% is nothing special. I just think of 100% as “about the time when I should start trying to kill the opponent instead of damage them”, though this varies depending on my character, the opponent’s weight, and the stage size.
However, there is something special about 100%. Except in Smash 4, most edge recovery actions become slower when your character has 100% damage or more. Climbing onto the stage, jumping up, rolling onto the stage, and edge attacking are all affected. Those actions’ animations are different, too. Roll distance changes. And attacks are different, slower but usually stronger. (The action of letting go of the ledge is unchanged).
For example, when Donkey Kong performs an edge attack at under 100% damage, he does a flying backwards leap onto the stage and hits enemies with his butt. This places him rather far away from the edge. When Donkey Kong edge attacks at 100% damage or more, he slowly pulls himself up onto the stage, close to the edge, and slaps sideways. In SSB, this slapping attack does the same amount of damage as the backwards slam attack, but in SSBB, the slap does more damage than the slam.
In Super Smash Bros. 4, this ledge-action-speed mechanic does not exist, but a different mechanic related to 100% damage is added. The Fairy Bottle item will only heal whoever holds it or whoever it hits when that player has 100% damage or more. Otherwise, the Bottle acts as a throwing item. (Like the Heart Container, the Fairy Bottle heals 100% worth of damage.)
There are other significant exact percentage numbers apart from 100%:
Source: “Damage” on SmashWiki. I can confirm at least the edge recovery differences from having noticed them in-game myself.
It's not literally a percentage of anything; it's just a number that determines how responsive your character will be to a hit. The higher the number, the farther you fly when someone hits you, up to a maximum value of 999.