Damage Down is a detrimental status effect which reduces the damage dealt by the affected player. It is received after being hit by an avoidable boss mechanic or if one or more people do not perform certain mechanics correctly (e.g., missing a tower soak).
The strength and duration of the effect depends on the duty. For example, damage down effects in ultimate raids are much more potent and longer-lasting (-50% for usually 3 minutes) than in a first-floor savage raid (-25% for 30 seconds). It can sometimes be a stacking debuff, with a max stack of 16. In newer duties, this debuff does not stack.
Players will usually only receive this or Vulnerability Up in a given duty; rarely both.