Absorption is the process of using a particular down special move to "capture" an "energy-based" projectile (or just a general hitbox in rare cases). 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 added some explosive-type attacks that were not able to be absorbed in previous games, yet did not include all of them.
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 the likes of Villager's 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.
As of version 1.1.3 in Smash 4, absorbed attack power from fellow teammates during Team Battles with Friendly Fire turned on is halved. This essentially hindered the effectiveness of moves with absorption properties in doubles as they were initially quite potent in Brawl and earlier versions of Smash 4.
List of moves with absorption properties
List of absorbable attacks