Short answers, pulled from the story.
Plato's Republic is a Socratic dialogue, written around 375 BC, concerning justice, the order and character of the just city-state, and the just man. In it Socrates debates whether the just man is happier than the unjust man, and proposes an ideal city called Kallipolis ruled by philosopher-kings.
The Republic centers on Socrates, who debates with various Athenians and foreigners. Key figures include Glaucon and Adeimantus, who press him to defend justice, along with Cephalus, his son Polemarchus, and Thrasymachus, who each offer definitions of justice that Socrates refutes in Book One.
The allegory of the cave appears in Book Seven of the Republic and depicts Plato's distinction between the world of appearances and the real world of the Forms. It builds on Socrates's two earlier analogies, the Sun and the Divided Line, which illustrate the Form of the Good.
The noble lie, or gennaion pseudos, is a myth the rulers of Kallipolis perpetuate so citizens accept their roles. In the myth of the metals, each person is said to contain gold, silver, or bronze, suiting them to rule, guard, or produce. Socrates claims believing it would make people more inclined to care for the state and one another.
Plato's Republic argues that only philosophers should rule because they have knowledge through their understanding of the Forms, while others have mere opinion. Socrates concludes that philosophers are the most just and least susceptible to the corruption of greed and power, and that the philosopher-king owns no property and receives no salary.
Plato's Republic shaped works from Cicero's De re publica and Augustine's The City of God to Thomas More's Utopia and Ibn Rushd's commentary. Mussolini admired it while Karl Popper attacked it as totalitarian in his 1945 book The Open Society and Its Enemies. In 2001 a survey of over 1,000 academics and students voted it the greatest philosophical text ever written.