Dueling is a is PvP contest of skill, class and/or equipment between two members of the same faction (who could not normally attack each other). Basically, it allows one player to challenge another otherwise allied player to a duel, and if the proposed opponent accepts, the two will be allowed to fight each other for a preset amount of time. In most games, dueling does not allow one player to kill the other, but merely to incapacitate and thus win the duel. Equally, there is generally no rewards to be gained from dueling other than the satisfaction of victory.
The exact mechanics in WAR have not been revealed, but it has been confirmed that dueling will be available in some form by Marc Jacobs.