What are the two meanings of the Judgement of Paris?
You would need to be more specific as to what you mean by "two meanings" for one to answer fully. That is to say, there is only one "Judgement of Paris"--a myth about events that led to the start of the Trojan War, often represented in classical art as it provided painters with the opportunity to paint three popular beautiful goddesses and the famous Prince Paris of Troy all in the same picture. Now, the word "meanings" is ambiguous: How does any story contain meaning? Sometimes, the story seems to point to an obvious moral or lesson, or we could say the story has a theme. If many people agree about this "meaning" then the story could be said to have a meaning. If people see more than one meaning, and can provide evidence and a rational train of thought to support those alternate meanings, then the story could have two, or three, or many more meanings. I wonder if your teacher (if this comes from a homework question) has taught two interpretations of the events in the story and is asking to see whether or not you listened to the class discussion. However, I know of no official second meaning. So I will quickly review the basic meaning as I see it, then speculate briefly about possible extensions or implications.
The Judgement of Paris refers to the myth where Eris, the goddess of discord, was not invited to the marriage of King Peleus to the sea nymph Thetis (parents of the Greek hero Achilles who is the star of the Iliad). To avenge herself for having been excluded, Eris tossed a golden apple inscribed "To the Fairest" into the festivities. Three rival goddesses each claimed the apple was obviously intended for herself: Zeus' wife Hera, Zeus' favorite daughter Athena, and the goddess of Love--Aphrodite. These goddesses asked Zeus to say who the apple really belonged to, but he was too wise to get involved. However, Zeus recommended an alternate judge, Prince Paris of Troy, who was living as a shepherd. Paris's father, King Priam, had sent his son away from Troy because Paris was fated to destroy his country.
Well, the goddesses not only went to see Paris; they also each attempted to bribe him! Hera, as Queen of the gods, offered him worldly power over men. Athena, as goddess of wisdom, offered him wisdom. And Aphrodite, as goddess of Love, offered him, well, the love of the most beautiful woman on earth. This beauty contest, as mentioned above, is frequently painted. Well, being a man (and this is where "meaning" comes into it), Paris chose Aphrodite. He wanted the love of the most beautiful woman on earth.
So, what is the meaning? Well, if you accept the idea that men "think" with their sexual organs, then that is one meaning. Offered power, which many men desire; or wisdom, which many men desire; Paris chose Love or Sex, things many men desire as well.
The outcome of this myth is tragic. The most beautiful woman in the world was Helen of Troy, who was the wife of a Greek King named Menelaus. Menelaus' brother was Agamemnon, the most powerful of the Greek Kings. Paris visited Menelaus, stole the heart of Helen with the help of the goddess of Love, and Helen ran away with Paris back to Troy. And the Greeks all went to war against Troy over this affront.
Is there a meaning beyond "men desire love more than wisdom or power"? Perhaps if we extend the meaning of the word "judgement" to include earning the fate you deserve. Troy was destroyed as a result of Paris's choice. After all, he had been fated to destroy his country. Having your country destroyed, your father and brother killed, over your choice to steal another man's wife could be seen as a form of judgement. We could say that History has not judged Paris well: Where his brother Hector is the great Trojan hero of the Iliad, Paris is presented as a coward and weakling, good only at love. We could say that the lesson of the myth is that from small decisions come earth-shattering results. Again, I would need to be a mind reader to know what your teacher refers to when saying there are "two" meanings to this myth.