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, stylistically, via the concept 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 despised democracy and uses "The Republic" to point out some of its weaker points: susceptibility to demagogues, rule by unfit "barbarians" etc. Arguably, the term "democracy" was used in antiquity differently, for mass rule that had gone haywire, although there are sources from Plato's time that indicate that the meaning was not really that different.
The ideal city as depicted in The Republic should be governed by so-called philosopher-kings as the only ones to be trusted to rule, since because of the human tendency to corruption by power and thus tyranny, ruling should only be left to those who would rather do something else (in this case, philosophize). Plato does not tell us much about the lower classes of society, he is only interested in the ruling class. For them, there should be no individual possessions, nothing to make them "weak", like spices or sweets. They must not mix with the lower classes, and they are forbidden to criticise the laws, until they are old, and then only if no youngsters are present. There are no families, no child may know its parents, the parents may not know who are their children. The rulers assemble couples for reproduction, based on breeding criteria. The individuals affected by this decision must comply. Sex for pleasure is fine, but if children are produced this way thay must be killed. As Plato recognised the destabilising effects of population growth (which ultimately led to Sparta's demise), it was important to him to keep the number of inhabitants constant.
The city of the Republic has struck many modern critics as unduly harsh, rigid, and unfree; indeed, as a kind of prequel to modern totalitarianism. Karl Popper is perhaps the today still best-known protagonist of that view, which is the view generally represented in modern introductory college textbooks on political philosophy.
At the time Karl Popper wrote his critique, 1945, Plato was usually seen in a much more favorable light, for example by Hans-Georg Gadamer in his 1934 classic, Plato und die Dichter (and several other works), in which the city of the Politeia is seen as a heuristic utopia which should not be pursued or even be used as an orientation-point for political development. Rather, its purpose is said to be to show how things would have to be connected, and how one thing would lead to another - often with highly problematic results - if one would opt for certain principles and carry them through rigorously. This interpretation argues that large passages in Platos writing are ironic (for which to detect, of course, an unusually high-level of proficiency in ancient Greek is required). Hence, Plato's entire oeuvre would be much less totalitarian, but only a demonstration where rigorous thinking starting from wrong premises can get you.
One of the most convincing arguments against this interpretation (even for those whose ancient greek is pretty bad) is that Plato's academy has produced a whole lot of tyrants. Following to the "Plato-was-just-being-ironic"-Hypothesis, many of Plato's own students were not able to recognise his subtle irony, although their proficiency in greek cannot be disputed (and they could talk to him in person to clarify things). Among his direct students where Klearchos, tyrant of Heraklia, Chairon, tyrant of Pellene, Eurostatos and Choriskos, tyrants of Skepsis, Hermias, tyrant of Atarneos and Assos, and Kallipos, tyrant of Syracuse.
In this book, Plato also introduces his theory of forms or ideals concerning the nature of reality. The world around us is likened to the shadows cast onto the wall of a cave.