Crouching
Crouching (sometimes refered to as ducking and internally labeled Squat in Super Smash Bros. Melee's debug menu and in Super Smash Bros. Brawl's files), is a ground-based position achieved by holding down on the Control Stick (D-pad on the standalone Wii Remote) when standing. Crouching shortens the character's hurtbox, allowing them to dodge some high-profile attacks and projectiles, and can be used to crouch cancel enemy attacks. From Brawl onward, a crouching character can also crawl if they have the ability. In Melee, crouching is also used often to cancel dashes. The crouch can then be stopped by simply stopping holding down on the Control Stick.
The utility of a crouch can help a character dodge attacks is highly dependent on the crouching animation and the character's overall size. Some characters can become very difficult to hit since they essentially hug the ground (such as Kirby, Jigglypuff, Snake, Sheik and Wii Fit Trainer). Other characters, however, barely lower themselves at all by crouching (such as Meta Knight, Bowser and Pac-Man), making their crouches extremely limited in use. Other characters like Donkey Kong and Bowser are so large that crouching barely makes a difference in their ability to dodge. All characters possess unique idle animations while crouching that replay constantly (such as Mario looking around, or the Wii Fit Trainer doing push-ups).
Characters can activate down tilt, down special and down smash while crouching, allowing for some niche techniques in competitive play. Kazuya specifically has several attacks that can only be activated by already being in a crouching position. Also, repeatedly crouching in rapid succession is refered to as "teabagging" and is often done intentionally by players as a pseudo-taunt to disrespect and frustrate opponents.
Super Smash Bros. for Wii U digital manual description
"Tilt down to crouch and reduce the amount of knockback that results from enemy attacks."
Gallery
Mario crouching in Melee.
King Dedede's crouch in Ultimate.