There's no way of understating the influence Plato's The Republic had on the history of Western thought. Whitehead said that all philosophy after Plato was nothing but a footnote to what Plato already said (he wrote several dialogues).
Plato was one of the first to start off the great "utopian" tradition of writing about a perfect world, a perfect society, the harmonious family, the best City, a sublime life-after-death, a tranquil existence within oneself--all imaginations that could be real, if only we tried it this way. Everyone has thought of their own version. Think of Jesus' "kingdom of God" and St. Augustine's "city of God" and Thomas More's "utopia" (called "utopia" as a satire because in latin it means "no place") and Martin Luther King's "beloved" community" and B.F. Skinner's in Walden Two. I met a guy at the library that was actually part of a real utopian commune in the U.S. (Emerson almost joined "Brook Farm"). Check out this book for a history of Utopias.
Plato imagines that the perfect city is a mirror of the perfect person. People fundamentally have three parts to their Soul: the rational part, the "spiritive" part (as in a warrior has spirit), and the "appetite" part (or desires). A City has the same parts, and they rank in the same order. Philosopher-Kings rule the city as model's of rational thinking, warriors protect it as model's of spirit and courage and braveness of heart, and the "commoners" make and trade all the stuff as model's of drive and desire and want. Thus we have a city as an extension of the perfectly organized self.
Plato and the gang, very regretfully, decide not to let poets into his city. Read it to find out why!