To celebrate the marriage of Achilles' parents - Peleus and Thetis - it is believed Zeus threw a grand banquet, inviting all of heaven. That is, all except for Eris, Goddess of Discord.
Eris was rightfully annoyed, and dramatically crashed the party. She presented a large golden apple as a gift for "the one most fairest" among all guests. Athena, Hera, and Aphrodite each declared herself as fairest, turning to Zeus for a tie-breaking opinion; Zeus, realising the danger of favouring one over the others, deferred his vote to the Trojan mortal, Paris.
Paris met the three goddesses at Mount Ida, where each in turn attempted to bribe him with their divine favour. Ultimately, Paris awarded the apple to Aphrodite; in return, the goddess gave him the mortal world's most beautiful woman: Helen of Sparta, then wife of King Menelaus.
The Spartans did not appreciate this one bit, and set out to reclaim Helen - supposedly sparking the legendary Trojan War.