The Duelist Kingdom dinner party is a event that takes place after the exhibition duel between Kaiba and Pegasus. Here, it is announced that the duelists facing each other in the finals are chosen by random.
In the anime, the Duelist Kingdom dinner party takes place during Episode 28 of the first season.
Each duelist is given a letter, sealed in a fake replica of the Millenium Eye, which were placed inside the four finalist's soups. These would randomly decide each duelist's placement in the finals. Joey received "D", Mai received "B" and Yugi received "A". In the manga, Bandit Keith had not attended and was given "C" by default.
It was also announced that in order to win the prize of three million dollars or the right to challenge Pegasus to a duel, each duelist would require either "Glory Of The King's Hand" or "Glory Of The King's Opposite Hand" cards. One of either of these cards would also be required in each of their duels in order to qualify.
Joey is upset, as he has neither card, but Yugi gives him his "Glory Of The King's Hand"card in order to win the three million for his sister, and double their chances of one of them facing Pegasus, as he already possessed "Glory Of The King's Opposite Hand".