In order to make the selection of champions completely fair, the decision is left up to a very powerful magic artifact, the Goblet of Fire. The Goblet is contained in a ancient wooden chest which is encrusted with jewels. When the casket it tapped three times with a wand, it opens. The Goblet is then removed and set on top of the chest.
Dumbledore described the selection process in this way:
"Anybody wishing to submit themselves as champion must write their name and school clearly upon a slip of parchment and drop it into the goblet. Aspiring champions have twenty-four hours in which to put their names forward. Tomorrow night, Halloween, the goblet will return the names of the three it has judged most worthy to represent their schools. The goblet will be placed in the entrance hall tonight, where it will be freely accessible to all those wishing to compete. To ensure that no underage student yields to temptation, I will be drawing an Age Line around the Goblet of Fire once it has been placed in the entrance hall. Nobody under the age of seventeen will be able to cross this line...I wish to impress upon any of you wishing to compete that this tournament is not to be entered into lightly. Once a champion has been selected by the Goblet of Fire, he or she is obliged to see the tournament through to the end. The placing of your name in the goblet constitutes a binding, magical contract. There can be no change of heart once you have become a champion. Please be very sure, therefore, that you are wholeheartedly prepared to play before you drop your name into the goblet." (GF16)
The next evening, the Goblet was taken into the Great Hall after the Halloween Feast. On cue, it disgorged a slip of paper for each chosen champion. To everyone's shock, the Goblet chose FOUR champions. Barty Crouch jr., in disguise as Moody, had used a Confundus Charm on the Goblet to trick it into accepting a name from another school. Once the names had been chosen, the Goblet's flames died out and would not reignite until the next Tournament.
The description in the books is of a "roughly-hewn wooden cup" which is "entirely unremarkable" except for the blue flames coming out of it. It is kept in a wooden casket. The Goblet of Fire as seen in the film version only vaguely matches this description.