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 (the 'forms')--among other things. The work is also famous for its literary style: the text is presented as a discussion between Socrates and several other students at a dinner, discussing the nature of justice.
The title "Republic" is derived from the Latin title given to the work by Cicero. Plato's Greek language title, Politeia, described 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: susceptibility to demagogues, rule by unfit "barbarians" etc.
Plato believes that the ideal city should be governed by so-called philosopher-kings whom he felt were the only ones with enough wisdom to be trusted to rule. Since Plato regarded the typical individual as being dominated by his appetites and base instincts, he regarded democracy as among the worst forms of government (and only slightly better than tyranny).