Zeus, the ruler of Mount Olympus and king of the gods; god of sky, weather, thunder, lightning, law, order, and fate; had held a banquet in celebration of Peleus and Thetis' marriage, them being the parents of Achilles. Eris, however, was not invited since she would make the party unpleasant for everyone - being the goddess of discord.
Eris was angered by the rude snub, of course, and arrived at the party with a golden apple from the Garden of the Hesperides, the word "Kallisti" inscripted into it, meaning "for the fairest". Three goddesses claimed the apple, they were: Hera (queen of marriage, women, childbirth, heirs, kings, and empires); Athena (goddess of wisdom, warfare, battle strategy, heroic endeavour, handicrafts, and reason); and Aphrodite (goddess of love, beauty, and desire). Zeus, reluctant to favor any of them as the fairest one, chose Paris (Trojan mortal) to judge the cases.
Overall the story ends with the beginning of the Trojan war.
For more (and better) information on this topic, check out Wikipedia's "Judgement of Paris" page at: http://en.wikipedia.org/wiki/Judgement_of_Paris_(mythology
I do not claim any of this information as my own.