Socrates is the narrator of The Republic, the central consciousness through whom everything flows and is filtered. He is also the protagonist of the text, if such a term can be made to apply. Above all the text is the first person record of a philosophical dialogue, an inquiry into the question of justice; hence there is very little traditional action and movement. It is an intellectual pilgrimage. We come to understand Socrates principally through his mind, which appears at first meandering though undoubtedly shrewd. That he "knows nothing" seems to clearly underscore a titanic intelligence. But he scarcely does anything aside from converse. As we learn from one of his auditors at the beginning of the dialogue, it is precisely for his provocative and illuminating conversation that he is famous.
Gradually, a pattern emerges from the constant testing and re-testing of ideas submitted by his auditors. Socrates will not introduce an idea on his own; his provisional conclusions, irrefutably his own, are derived or originate in the offerings (most often, the erroneous offerings) of his auditors. What Socrates' really knows is incommunicable, but he can invariably tell when someone else fails to recognize his own ignorance. By the end of The Republic, however, it is possible to say what Socrates' more or less believes. He believes in the four virtues: courage, wisdom, temperance, and justice. He believes philosophy is man's most noble, as well as most useful, occupation; in fact, it is his obligation. He appreciates though distrusts poetry. And he is extraordinarily humble and patient, never rejecting an idea without a fair inquisition. Socrates' character, apropos of his moralitywhich requires that one actually live itmaterializes through his thoughts. There can be no doubt that when he speaks of the strenuous education of the guardians, he considers himself a guardian. But most importantly perhaps, Socrates' philosophical speculations embody a process rather than a philosophy. He has no doctrine, no dogma, no allegiance other than to the Truth. That is, Socrates' method is in accord with the nature of inquiry and of intellectual exploration itself: he is his style; he is the dialectic, leaping, as he explains, from one invisible step to the next in pursuit of the good.
Glaucon is the name of one of Plato's older brother and, in The Republic, remains Socrates' closest and most loyal disciple. Throughout the dialogue, he never leaves his master's side. In Book II, after the confrontation with Thrasymachus, Glaucon agrees for the sake of argument to oppose Socrates. He is fairly good at presenting conventional positions and so, without injustice, he may be considered the embodiment of conventional thought. At a later point, he is compared to an auxiliary in the State apparatus.
The second of Plato's brothers, Adeimantus is a source of poetry and literature in the course of the dialogue. He is also avowed disciple of Socrates and, like the others, declines when given opportunities to lead the discussion. In Book III he has difficulty understanding Socrates' idea about narrative style, which forces the philosopher to clarify a sophisticated point.
The elderly father of Polermarchus, Cephalus' endearing ideas about old age initiate Socrates' musings on the nature of the virtuous life. It is in his home that the dialogue commences. Cephalus, to the extent that he believes the lamentations of the aged are the result of their defective characters, represents the fruition and logical conclusion of right action. In spite of the physical discomfort of late maturity, Cephalus is happy, and this, after all, is Socrates' point.
Polermarchus invites Socrates to his home, eager for the conversation. The son of Cephalus, Polermarchus cherishes very common ideas. In Book I he suggests that justice is giving each man that which he deserves. And in Book V, after being caught whispering, he accuses Socrates of laziness and demands the philosopher explain in detail the mechanism of family and community in his State.
Thrasymachus is the fierce embodiment of tyranny. He explosively interrupts the dialogue in Book I but refuses to state his position without being paid first. Socrates' disciples pay him and Thrasymachus gives his definition of justice: the interest of the strongera concise statement of tyranny. The repercussions of his argument are used by Socrates throughout the discourse to develop a better idea of perfect injustice.
The Republic Questions and Answers
The Question and Answer section for The Republic is a great
resource to ask questions, find answers, and discuss the novel.
There is too much information for me to share it all here. Check out the link below, I think it will help you formulate your answer. If you have any more specific questions, feel free to come back and ask them here.