Short answers, pulled from the story.
The Judgement of Paris is a myth in which the Trojan prince Paris was asked to choose the most beautiful of three goddesses: Hera, Athena, and Aphrodite. He awarded a golden apple inscribed "To the fairest one" to Aphrodite, who had bribed him with Helen of Sparta, the most beautiful mortal woman in the world. The episode is considered the instrumental cause of the Trojan War.
Eris, the goddess of discord, was excluded from the wedding of Peleus and Thetis because it was believed she would make the celebration unpleasant for everyone. In revenge, she arrived uninvited and threw a golden apple inscribed with the words "To the fairest one" into the proceedings.
Hera offered to make Paris king of Europe and Asia. Athena offered wisdom and skill in war. Aphrodite offered the most beautiful woman in the world, Helen of Sparta, wife of the Greek king Menelaus. Paris accepted Aphrodite's bribe and awarded her the apple.
The Apple of Discord is the golden apple Eris threw into the wedding of Peleus and Thetis. The inscription on it, kallisti or kallistei, is the Greek word meaning "to the fairest one," the dative singular of the feminine superlative of kalos, meaning beautiful. The apple and the word kallisti later became a principal symbol of Discordianism, a post-modernist religion.
Lucas Cranach the Elder is said to have painted the Judgement of Paris approximately 23 times, making it one of the most repeated subjects in his body of work. The subject was especially popular among Northern Mannerist painters.
A brief allusion to the Judgement of Paris appears in Homer's Iliad at lines 24.25-30, dating to around the 8th century BC, indicating the story was already familiar to audiences at that time. A fuller telling appeared in the Cypria, a now-lost work of the Epic Cycle that survives only in fragments and a summary.