The Republic is perhaps Plato's best-known dialogue and one of his most influential. In it he explains, through the character of Socrates, the fundamentals of his political philosophy (presented via the conceit of a Utopia), his ethics, and his theory of universals--among other things.
The title "Republic" is derived from the Latin title given to the work by Cicero. Plato's Greek langague title, Politeia, refers to the government of a Polis or city-state. The character Socrates and his friends discuss the nature of an ideal city rather than the nature of the Athenian democracy.
The Republic, bears little to no resemblance to the modern political institution that we, in modern times, know as the Republic. Plato despises democracy and uses "The Republic" to point out some of its weaker points: succeptibility to demagogues, rule by unfit "barbarians" etc.