Paris (also known as Alexander), son of Priam, king of Troy, appears in a number of Greek legends. Probably the most well known was his abduction of, or elopement with Helen, queen of Sparta, this being one of the immediate causes of the Trojan war. Later in the war, he fatally wounds Achilles in the heel with an arrow.
When Paris was born, an oracle claimed he would be the downfall of Troy. His parents sent him to Mount Ida, hoping he would stay out of trouble there.
The goddesses Hera, Athena and Aphrodite had been invited along with the rest of Olympus to the forced wedding of Peleus and Thetis, who would become the parents of Achilles, but Eris (the goddess of strife) had been snubbed because of her troublemaking inclinations.
Eris therefore tossed into the party a golden apple inscibed "Kallisti" -- "For the most beautiful one"-- provoking the goddesses to begin quarreling about the appropriate recipient. The hapless Paris, Prince of Troy, then living on Mount Ida, was appointed to select the most beautiful. Greek mythological morality being what it was, each of the three goddesses immediately attempted to bribe Paris to choose her. Hera offered political power and control of all of Asia, Athena skill in battle, wisdom and the abilities of the greatest warriors, and Aphrodite the most beautiful woman in the world, Helen, wife of Menelaus of Sparta. Paris was a red-blooded young man, and while the length of time he meditated on this problem is not recorded, he did eventually award the apple to Aphrodite.
Since Helen was the most beautiful woman in the world, she had had many suitors. To keep the peace between them, Odysseus made them all promise to defend the marriage of Helen and whomever she chose. She chose Menelaus and when Paris kidnapped her, arriving on a boat built by Phereclus, (according to some, she fell in love with Paris and left willingly), all of Greece attacked Troy -- the Trojan War.
For more information, see Judgement of Paris.
Oenone, Paris' first wife, was a nymph from Mount Ida in Phrygia. Her father was Cebren, a river-god. Paris kidnapped her and they were wed; Oenone gave birth to Corythus. When Paris abandoned her for Helen, she predicted the disastrous results of Paris' attempt at Helen (the Trojan War, Paris' death).
Alternatively, when Paris was mortally wounded he asked for her to heal him since she was known as a healer. Oenone refused and Paris died; she threw herself onto his burning funeral pyre.
A third possibility sometimes cited was her attempts at breaking up Paris and Helen. She sent Corythus to drive a rift between Paris and Helen but Paris didn't recognize his son and killed him.
There is an icon showing Paris presenting an apple to (one of) the Goddesses Athena, Aphrodite, and Hera. This icon is most commonly interpreted as the "Judgement of Paris", the assumption being that he is deciding which of the three is the most beautiful. An alternative interpretation is that Paris is shown receiving the gift of life from the trio of Goddesses. This interpretation is supported by the contention that the three goddesses represent a single goddess in her three aspects of nymph, mother and crone and that to choose the most beautiful makes no sense in this context.