Probably one of the most popular words to come out of Avatar: The Last Airbender, it translates to "fire meeting."
Agni Kai's are a popular (and violent) way of settling arguments in the Fire Nation. One person challenges another and they both face off in a fire-bending battle.
This was also one of the events that led of up to the banishment of Prince Zuko in the series. It also brings down his mentally unstable sister, Princess Azula.
"Bitch just stole my Fire Bits! I challenge you to an Agni Kai!"