Damage is a statistic on all weapons. It simply shows how powerful a weapon is; if a weapon has more power, each bullet/explosion/strike will remove more HP from the player's target.
Damage is affected by the location a bullet or melee strike hits; should the attack hit the torso, full damage will be done; should it hit the arms or legs, around 10% damage will be subtracted; should the attack hit the head, around 5 times the damage will be done, the enemy is usually killed instantly. (unless they have protection from Head gear or unless the weapon isn't strong enough and sometimes shot from a great distance)
If Friendly Fire is on, and the player shoots or attacks an ally, the full damage will not be done. Approximately half the damage that would be done to an enemy attacked in the same way will be done to the ally, unless the mode is Spy Hunt, which does full damage to allies.
Explosive damage is not affected by limbs.