The Republic is one of the most significant works in the field of political science. The original name is Politeia. The author, the Ancient Greek philosopher Plato, published it around the years 375 to 370 BC. This classical work consists of ten books. Besides, it is his second-longest work after Laws.
It falls into the second period of Plato. That was the period of his true idealism. Further, as usual with Plato, he structures the books as a dialogue. The lead of the conversation is Socrates. He addresses politics and justice. Besides, the fall of Athenian democracy and the Peloponnese wars influenced the work.
Socrates, the protagonist, asks a group of Athenians what justice is. After they discuss it, he defines a perfect and just city-state. It stands on education and loyalty. Further, he introduces three classes. The producers are the largest class. Besides the army and rulers, everyone else is in the class. Secondly, the auxiliaries protect the city. Finally, guardians are the rulers. He calls them the philosopher-king because they are the wisest.
In summary, the best leader is the one who has the most knowledge. He pictures it on the Allegory of the cave. People live in a cave where they can only see shadows. Then, one manages to escape and perceives the reality on the outside. However, he then returns to the cave to teach the others.
Further in The Republic, through Socrates, Plato discusses the forms of government. These are in order from best to worse: aristocracy, timocracy, oligarchy, democracy, and tyranny. Thus, the aristocracy is a just government with the philosopher-king in the rule. However, it slowly degrades down to tyranny.
Plato’s description of democracy resembles anarchy. He feels that people have too much freedom, and on the other hand, no unity. In a democracy, an ill-minded person can buy people’s trust with a gift. Then, the system degenerates into tyranny. However, the Athenian democracy was a direct form rather than representative. For example, James Madison has comparable views in The Federalist.
In the context of Plato's Republic, the word "republic" means constitution rather than the form of government. See more here: Politeia