A Duel is a simple 1 on 1 battle between you and another lord. Your equipment (and the rival lord's equipment) will be limited to Tournament Weapons, which means one or two hits will conclude the battle. So be quick and cautious.
If you force a lord to relinquish his suit to a lady or denounce him openly, there is high possibility that he will challenge you to duel. This often results in a relationship loss with the lady as well.
In Warband, it replaces Trial of arms. The aim of making the duel system was to give player more challenge (by restraining one to standardized equipment). However, duels are much easier than their predecessors due to some tactics the AI cannot counter (e.g. blocking the first hit and continuously kicking the opponent until he is knocked down).
Duel mode in multiplayer is a system for combat where opponents will square off 1v1, and a challenge has to be made and accepted before a fight can start. It is far more orderly and less hectic than the other multilayer modes, but can still be enjoyable if your goal is to watch how skilled someone is at combat in the game. Just as with other multiplayer modes, you may buy equipment and pick a starting class.
To start a duel in multiplayer mode walk up to a player and press the action key. If they accept (by pressing the action key themselves), there will be a countdown from three, and then the duel starts.