Glaucon is Socrates's best bud. We know he's special because he and Socrates were checking out this religious festival together, and Socrates doesn't just hang with anybody. Now, since Plato's dialogues, especially the Republic, aren't really about telling a story and are more about describing a conversation, we don't get a whole lot of character development. In fact, some critics have even suggested that Socrates's interlocutors aren't even really "characters."
Whether we think that's true or not, we still get some little hints about what differentiates these people other than their names. Glaucon seems to be the kind of guy who's totally on Socrates's side... but he just wants to be really, really sure of things. This is why he brings up the whole ring of invisibility story in the beginning: "For Socrates," he says by way of an explanation, "though [the idea that an unjust man is better off is] not at all my own opinion... the argument on behalf of justice... I've yet to hear from anyone as I want. I want to hear it extolled all by itself... That's the reason why I'll speak in vehement praise of the unjust life..." (358c-d).
Glaucon wants to play the devil's advocate in order to get all the deets that Socrates might leave out if he were just spouting off his arguments to a captive audience. It's because of Glaucon that we get to see exactly how Socrates gets from point A to B to C. It's a good thing this dude is around; otherwise, we might be totally lost.
Thrasymachus might be the most memorable character in Plato's Republic, but maybe not for the best reasons. This guy has a serious temper, and he finds Socrates really annoying.
But beyond just throwing some fits, Thrasymachus actually offers some pretty valuable challenges to Socrates's whole method. "If you truly want to know what the just is," he yells, "don't only ask and gratify your love of honor by refuting whatever someone answers—you know that it is easier to ask than to answer—but answer yourself and say what you assert the just to be." (336c).
Thrasymachus might have a point. The Athenian government that sentenced Socrates to death certainly seemed to agree. There can be something profoundly frustrating about Socrates's method of acquiring knowledge, since it rarely involves him actually answering anything... and it usually involves making his interlocutors feel inadequate. While reading Socrates's interrogations, we definitely get the impression that this dude knows a lot more than he's letting on, and that can be totally frustrating.
Polemarchus is the man of the house, and he's a guy who just doesn't take no for an answer. When Socrates suggests that he might want to head home, Polemarchus says, "Well then either prove stronger than these men or stay here" (327c). Not the spirit of compromise, right? Polemarchus is eager to get in on the philosophical discussion right at the beginning, but he takes more of a side role for the majority of the debate.
Even though we get absolutely zero hint of this from the dialogue itself, Polemarchus was actually a historical person involved in a rebellion against some Athenian tyrants. Check out our "Setting" section for more.
Although Cephalus is only briefly featured in the dialogue, you could say he's the one who starts it all. It's his conversation with Socrates about old age that gets the whole ball rolling, and it's his perceptive unwillingness to accept what everyone says about old age that shows that he has a bit of a Socratic inclination himself. Instead of seeing old age negatively, he says that "...in every way, old age brings great peace and freedom..." (329c). Rock on, Cephalus.
Adeimantus is a major part of the debating that goes on in the Republic. Most of the big realizations and discoveries go down while Socrates is talking to either him or Glaucon. Unlike Glaucon, though, Adeimantus seems to be less interested in challenging what Socrates is saying. He's more of a go-with-the-flow kind of guy.