Crouching (or Squat internally in Super Smash Bros. Brawl's files and in Super Smash Bros. Melee's debug menu), often simply called ducking, 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 damageable collision bubble, allowing them to dodge some 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.
How well a crouch can help a character dodge attacks is highly dependent on the crouching animation. 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. Similar to idle poses, all characters present unique animations while crouching that replay constantly (such as Mario and Donkey Kong looking around, or the Wii Fit Trainer doing push-ups).
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.
Mario holding his crouch in Melee.
King Dedede's crouch in Ultimate.
