Tag or Tagging refers to causing initial damage to a MoB claiming it as yours.
The way the game mechanics works is, as soon as a player causes damage to a mob, no other player outside of that player's group can get credit for the kill or loot the corpse (they will be able to skin it after it's been looted, however) even if they do the majority of the damage. When a mob is tagged by someone else, the name plate of the mob will turn greyish indicating that the mob belongs to someone else.
This can be useful for people with high level characters helping them that are not in their party. The lower level tags the mobs and then, when the name plate goes grey, the higher level character finishes it off. Conversely, tagging can be used for nefarious purposes. The act of one player tagging a mob that another player is obviously going for but doesn't have the ability to quickly tag (such as paladins who have no ranged pulling ability) is called kill-stealing and is generally considered bad form.
Note that the key is initial damage. That means if you hit the mob with a non-damage dealing debuff such as Faerie Fire, the mob can still be tagged by another player. This can also be the case with DoTs. It's possible for a warlock to put a DoT or two on a mob and someone else to tag (kill steal) it before the first tick of damage. If you're in a highly competitive area such as the Elemental Plateau, it's a good idea to lead with something that will cause immediate damage rather than waiting for the tick and hoping the mob is still yours when it does.