At an important marriage of King Peleus and the sea nymph Thetis, all of the divinities were invited except the goddess Eris, who was known as the goddess of Discord. Even so, she threw a golden apple into the banqueting hall with the words For the Fairest.
Naturally, all of the goddesses wanted it, but in the end the choice was narrowed down to three: Aphrodite (goddess of sensual love and beauty), Hera (Juno, queen of heaven), and Pallas Athena (goddess of wisdom, arts, and warfare).
They asked Zeus, the chief god, to pick the one who should receive the golden apple, but he refused to have anything to do with the matter. He told them to go to Mount Ida, near Troy, where the young prince Paris, also known as Alexander, was taking care of his father's sheep, and who would be qualified to decide who should receive the apple.
The three goddesses made promises in order to influence his choice as to which one should be chosen as the fairest. Hera promised to make him Lord of Europe and Asia; Athena, that he would lead the Trojans to victory against the Greeks and lay Greece in ruins; and Aphrodite, that the fairest woman in all the world should be his.
Paris gave Aphrodite the golden apple. This was the "Judgment of Paris", famed everywhere as the real reason why the Trojan War was fought.
But the goddess, Eris, who had not been invited [to the wedding of King Peleus and the sea nymph Thetis; (added for clarification)], was determined to put the divine guests at loggerheads, and while Hera, Athena, and Aphrodite were chatting amicably together, arm in arm, she rolled a golden apple at their feet. Peleus picked it up, and stood embarrassed by its inscription: "To the fairest!", not knowing which of the three might be intended. This apple was the protocatarctical cause of the Trojan war.
Special thanks to Brandon Chenault for providing the Robert Graves quotation.