Tapped is a state that occurs when a player damages a mob, locking its experience gain, loot and corpse to be available only to that player or anyone in that player's party. Another term for a tap is a "tag".
This state is indicated by the mob's health bar turning gray.
Only attacks and direct damage abilities are able to tap a mob; Damage over Time abilities and non-damaging debuffs will not prevent another player from tapping a mob.
Trying to use an item on a mob that has been tagged by someone else or someone from outside their party will return the error message "Target is tapped." However, players can still attack and kill the tapped mob even if they were not the ones to tap it.
It is possible for one player to tag multiple mobs in order to keep others from doing so. Although controversial, it is not against Blizzard's Rules of Conduct to tag multiple mobs.
Patch 3.0.8 (2009-01-20): All player spells which cause a creature to become aggressive to you will now also immediately cause the creature to be tapped.