Honor duels are a tradition that gained popularity first in the Oathlands. They were soon embraced by the Lyceum city-states, and eventually became the preferred method of dispute resolution across the Compact. In general, the populace considers honor duels a win all around: minor matters are resolved quickly rather than festering into greater conflict, and the general populace gets a show to watch. The showmanship aspect of it is part of why duels are nearly always fought by champions from the Champions Guild, rather than the aggrieved parties themselves.
Every duel must have an explicit dispute to resolve, a single reason that can be pointed to, and something to demand an apology over. It does not have to be a /good/ reason, but it must be explicit; "I don't like you" is not sufficient grounds for an honor duel. It is also not an official honor duel if a challenge is not formally issued. For purposes of the game, this means it must be posted to the Challenges board to be official.
If the challenged party declines the duel, it is considered a tacit admission that their position was wrong and should offer apology, and the challenger's claim is indeed accurate; the challenging party can safely claim the victory and move on with life. Otherwise, the victorious side is considered to have won the dispute.
Duels can be fought using many different forms of combat: in the Oathlands, duels are sometimes fought as jousts, while others might be wrestling, or the ever-popular swordfight. Terms for a duel are worked out beforehand: what type of combat it will be, what weapons and armor are allowed (such as material, but not insisting someone use a weapon they have no familiarity with and trying to stack an unfair fight), and what the victory condition is. For your average insult or slight, a duel to first blood is proper. For practical purposes, 'first blood' will be a specific amount of damage selected ahead of time by the parties involved; the first combatant to reach or exceed that amount of damage is considered bloodied and loses.
To kill one's opponent during an honor duel is considered a serious, nigh-unrecoverable stain upon one's honor, since the practice was adopted specifically to prevent escalating bloodshed over perceived -- or actual -- slights.
In duels for honor or duels for exhibition, such as a Rite to Gloria, it is appropriate to keep in mind the fight is for story, and realize that IC, no one would greet complaints of "but my opponent had better boots/axe/higher skill" with anything but scorn.
Note that trial by combat, while a right accused criminals can claim, is governed by a different set of rules.
(see also help files for "slights", "champions", "heirloom weapons", and "trial by combat")