Damage, or dmg, is a reduction in health or an action that reduces health via some sort of attack.
For example, a weapon attack or spell attack that reduces the health of the target is damage, but losing a health boosting buff is not considered damage.
Losing mana, or MP, from an attack can also be considered damage, but generally is not referred to in that way since reducing MP to zero will not cause death.
Note: A more accurate way to measure damage output is DPS.