What is "dual-boxing"?

Dual-boxing refers to running two clients at the same time (of the same game, usually) and controlling them via a single input (usually heavily macro'd). Multi-boxing is the general term applied to more than a single client at a time.

To "properly" dual-box in World of Warcraft, you need two separate accounts. The comment likening naming a dual-boxing character pair to an alt was misinformed.

Dual-Boxing is not forbidden by World of Warcraft's Terms of Service, though those who do the practice are generally disliked by other players, due to their greater presence and combined power level, and the intrinsic PvP nature of the game.