(written free from memory here, so dont sue me if I screw up on the details).
Zeus threw a big party at the Olympus, but he forgot to invite one of the gods. It was [insert correct name of female goddess here] and she was the god of jealousy or something like that.
Naturally this pissed her off royally, so she decided to ruin the big party. She manufactured an apple of pure gold, and on the apple she inscribed "to the most beautiful one", and she threw the apple into the palace.
When it landed on the table, one of the Gods picked it up, and read the inscription. And soon the worst cat fight in history broke loose. Every female goddess in that party decided that she was the most beautiful one and claimed the apple for herself. All the male Gods tried to avoid getting sucked in to that argument, and hid in a corner. Soon 3 major contestants emerged. Hera, who was Zeus wife, Athena, who was the goddess of war or hunting (cant remember which one), and Venus, the goddess of love. These three women approached Zeus with the apple and demanded that he decide which one would get the apple. Naturally (since he had half a brain) he refused to touch that one. Furthermore, he told the women to stop pestering the gods with the apple-issue.
The three women would not give up however, they still needed someone to judge who was the most beautiful one amongst them. Someone came up with the idea to let a mortal decide the issue.
Enter Paris.
He was the king of Troy, and he was casually strolling along a beach somewhere in Greece when -poof- Hera, Athena and Venus appeared beside him. They explained the issue, and demanded that he choose among them who was the most beautiful one. Since he was a mortal, he did not dare to refuse. But he asked for one hour to think about it alone. The three women agreed, and left him alone.
Naturally they all tried to bribe him though. First, Hera approached him and said that if he choose her, she would make him the wealthiest man in the world. Then Athena came and said that if he choose her, he would never loose a single battle in all his wars. And lastly, Venus came, and she said that if he choose her, she would give him the blessing of true, happy love.
After weighing his options Paris choose Venus. Something that seriously pissed of the other two contestants. Athena vowed that Paris would never win a war again, and Hera vowed that she would make sure he died a poor man. Venus was happy though, and promised Paris that he would be blessed with true and real love, and that people would always remember his and his lovers fate.
Sometime after this, Paris was invited to the wedding between [insert name of Greek king here] and Helena. Naturally he and Helena fell in love, since she was like the most beautiful girl ever, and Venus kept her promise. They fell in love, and decided to run away together, which they did. The Greek king was a little upset by this, understandable, since Paris ran away with his bride on the wedding night. War was declared. Enter the Illiad.