If the ability holder faints from a contact move, the attacker loses a quarter of its maximum health. Aftermath does not activate if a Pokémon with Damp is on the field.
If the Pokémon with this ability faints, it occasionally causes a blast similar to explosion, taking away half the health of teammates in its radius and 40 HP of hostile Pokémon.
All Pokémon that are able to have this ability can learn Explosion. Its name in Japanese is more along the lines of induced explosion which explains why Damp prevents Aftermath.