Avatar: The Last Airbender's Fire Nation has a long history of violence, even using it to solve internal conflicts to literally fight fire with fire. An Agni Kai, a Firebending duel for the sake of one's own honor, is a sacred tradition in the Fire Nation. When two Firebenders find themselves at odds with one another for any reason, they can opt to settle their dispute by engaging in an Agni Kai. But what do the words 'Agni' and 'Kai' actually mean, and how was the duel used in Avatar?
In Sanskrit, the word 'Agni' can be interpreted as meaning 'fire,' while 'Kai' is the Hawaiian word for 'sea.' "Sea of Fire" is the perfect way to describe the Agni Kai between Azula and Zuko in Airbender's series finale. Meant to be a measure of last resort, the victor of this confrontation is declared the winner of the feud. It's unclear if this applies to legal disputes though, as these duels are usually fought over pride rather than property.
Similar to Old West standoffs, two combatants stand with their backs to one another before the conflict begins. Higher-ranking members of the military will face each other before removing their ceremonial shoulder garments. Men are meant to be shirtless and barefoot, most likely in order to make it clearer -- and more painful -- when a burn is successfully inflicted. Though in the case of Zuko's and Zhao's Agni Kai, no burn was made as the banished prince accepted his opponent's yield.
Killing one's opponent in an Agni Kai isn't required, but it isn't exactly frowned upon either. Sparing an opponent was once a gesture of honor, but Fire Lord Sozin turned it into a sign of weakness. Typically, like in Zuko's case, an Agni Kai ends when one combatant burns another. The loser of the duel, if they survive, will shave portions of their head as a sign of their humility in defeat.
The Agni Kai is a long-revered tradition in Avatar's Fire Nation, with even the royal palace containing a chamber specifically for these duels. However, challenging someone to an Agni Kai on certain holidays is forbidden. It is considered disrespectful to the spirits to burn someone on a day of celebration. In more modern times, it has also become something of a spectator sport with large crowd turnouts. Sparky Sparky Boom Man aka Combustion Man earned his savage reputation through numerous victories in these battles.
The Fire Nation places a heavy emphasis on what they view as personal honor, and defeat in an Agni Kai means losing that honor. Zhao disgraced the tradition in his duel with Zuko by lashing out with flames after having been defeated. The act was so disrespectful that even Iroh stepped in to admonish the Fire Nation admiral for his revolting lack of honor. Azula also snubbed tradition by attacking Katara, despite being the one who challenged Zuko to an Agni Kai in the first place.
Zuko's reign as Fire Lord likely put an end to Agni Kai in Avatar. Aside from his being a victim of their cruelty, it is a violent tradition from an ugly era in Fire Nation history. Zuko himself stated that he intends to usher in a new era of love and peace, which probably won't include people burning and/or killing each other over honor.