Upon entering the Duel Hall you can challenge other Eternal Duel players for fun and bragging rights.
There are no permanent benefits from a duel, no experience or gold is gained, but sometimes it's fun to test your skill against others.
You can duel players up to 20 times per day by entering their ID number in the Duel Hall and testing yourself. Bare in mind that some players do not want 20 duels from the same person everyday, and can report you for duel spam. So spread them out a little.
You can also duel players by clicking on a player's name/profile and scrolling down to "Duel (player name)".