Short answers, pulled from the story.
The Myth of Er is a legend that concludes Plato's Republic at book ten, chapters 614-621. It follows a soldier named Er, son of Armenios, who dies in battle, revives on his funeral pyre twelve days later, and recounts a detailed vision of the afterlife, including the soul's cycle of death, judgment, and rebirth.
After death, souls are judged at a place with four openings and sent either upward through the sky or downward underground. The wicked pay a tenfold penalty for each wrong act committed in life. After seven days in a meadow, souls travel to the Spindle of Necessity, draw lottery tokens, and choose their next lives before drinking from the River Lethe and being reborn.
The Spindle of Necessity is Plato's model of the cosmos. It carries eight orbits corresponding to the fixed stars, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. The three Fates, daughters of Lady Necessity, keep the spindle's rims turning, linking cosmic order to the cycle of moral consequence governing souls.
Odysseus selects a quiet, private, ordinary existence after remembering the suffering caused by his love of reputation in his previous life. Plato uses this choice to illustrate the story's moral: those who have suffered from their own ambitions are more likely to choose wisely when given the opportunity to pick a new life.
Scholars have linked the two stories through the name of Er himself: he is identified as the son of Armenios, from Pamphylia, and his name echoes Ara. The scholar Armen Petrosyan argues that Plato's version reflects an earlier form of the Armenian legend in which Ara, like Er, rises from the dead after a battle.
In ancient Greek, the word mythos means word, speech, or account, not a fictional tale. Socrates uses the term at the end of the story to explain that because Er did not drink from the River of Forgetfulness, his account was preserved and could be passed on to the living.