Absorption is the process of using a particular down special move to capture a certain kind of projectile (or just a general hitbox in rare cases), usually described as "energy-based". It then provides a benefit to the character absorbing it based on the damage output of the absorbed attack. What constitutes an absorbable attack other than being labeled as energy-based is not consistent throughout the Smash series, and has no relation with move types or effects. Super Smash Bros. 4 allowed some, but not all explosive attacks to be absorbable when they were not in previous games.
Only four characters in the Super Smash Bros. series have special moves that can absorb specific attacks, namely Ness, Lucas, Mii Gunner, and Mr. Game & Watch. Ness and Lucas' absorbing special move is PSI Magnet, Mr. Game & Watch's is Oil Panic, and the Mii Gunner's is Absorbing Vortex. PSI Magnet and Absorbing Vortex heal the user based on the absorbed attack's power, while Oil Panic stores the strength of up to three projectiles for unleashing a single attack later. Absorption is not to be confused with Villager and Isabelle's neutral special, Pocket, or any other form of collecting items.
In The Subspace Emissary in Super Smash Bros. Brawl, Floows are also capable of absorbing attacks, though they do not appear to do anything with the absorbed energy. In Smash Run in Super Smash Bros. for 3DS, Chandelure appears to be capable of absorbing flame and darkness attacks to power up its own attacks, but absorbs all attacks with these effects, including non-projectile attacks.
In Super Smash Bros. 4 from version 1.1.3 onward, and Super Smash Bros. Ultimate, absorbed attacks from allies in team battles with team attack turned on have their damage reduced by half. For example, Ness' PSI Magnet, which increases the damage of absorbed projectiles by 1.6× in both games, decreases it instead by 0.8× for teammates' projectiles. This hinders the effectiveness of moves with absorption properties in doubles, as they were initially quite potent in earlier games/versions when combined with strong projectiles (such as Samus' fully charged Charge Shot, which could bring Oil Panic's attack close to one-hit KO potential).
List of moves with absorption properties
List of absorbable attacks