You can read Charmides online at the Perseus Project, as well as in French at Wikisource.
(in order of appearance)
Chaerephon was one of Socrates's oldest and closest friends. Plato presents him as a little bit crazy, in a good way, and comic poets like Aristophanes continually make fun of his bat-like/corpse-like appearance, so he and ugly-as-a-Silenus Socrates must have made a curious pair.
Critias was an older cousin of Plato and became one of the Thirty Tyrants.
Charmides was Plato's uncle and became one of the Thirty Tyrants.
Other young men
Socrates narrates how he and others have just returned from the Battle of Potidaea. Visiting the palaestra of Taureas, he finds a considerable number of people, among whom is his friend Chaerephon. Chaerephon asks about the battle, so Socrates sits with Chaerephon and Critias and answers questions. Socrates eventually redirects the discussion to philosophy and young men distinguished for wisdom or beauty or both. Critias, who sees a number of young men coming through the door, replies that he thinks the most handsome young man in Athens will soon be behind them: Charmides son of Glaucon, Critias's cousin. Charmides does in fact enter, and he is indeed very handsome. Socrates says that with such a handdsome face, if he has the perfect body that others say he does, he only needs one thing to be a man without equal: he has to have a well-formed soul.
Critias says that Charmides is exactly such a man. Charmides has been waking up with a headache lately, so he suggests that if Socrates wants to talk to him, he should pretend to have a remedy for it. Charmides comes over and Socrates offers a charm for the remedy, but notes that doctors don't just dispense medicines, but treat the body in such a way as not to focus only on specific problems. For the remedy to work properly, it has to be part of a larger treatment, which involves temperance. Critias says that Charmides excels in temperance, and Socrates asks whether this is so. Charmides, blushing, says that it's a hard question to answer, because it would be an odd thing to deny that you have it, but if you say you have it, you sound like you are boasting. So Socrates proposes that if temperance is present in Charmides, "it provides a sense of its presence," so he should be able to say something about what it is.
This then leads to an extended examination of definitions of temperance. Charmides first proposes that it is a kind of quietness, then, when that doesn't pan out, he suggests that it is a kind of shamefacedness or bashfulness. This also does not work. But Charmides then says that he heard someone claim that temperance is minding one's own business. Socrates suspects immediately that Charmides is parroting Critias, and he is unable to defend it. This brings Critias into the argument to defend his definition.
Critias in developing his definition specifies it in terms of the Delphic inscription: 'know thyself' and 'be temperate' are two ways of saying the same thing. Socrates presses this point about knowledge, and Critias commits to saying that temperance is a kind of knowledge of one's own knowledge. This turns out not to succeed. Thus the dialogue ends with perplexity about what temperance is.
It also ends on an ominous note, after a short consultation between Charmides and Critias, in which they conclude that Charmides should receive Socrates's charm remedy every day:
"This is the course I shall follow," he said," and I shall not give it up. I would be acting badly if I failed to obey my guardian and did not carry out your commands."
"Well then," said Critias, "these are my instructions."
"And I shall execute them," he said, "from this day forward."
"Look here," I said, "what are you two plotting?"
"Nothing," said Charmides--"our plotting is all done."
"Are you going to use force," I asked, "and don't I get a preliminary hearing?"
"We shall have to use force," said Charmides, "seeing that his fellow here has given me my orders. So you had better take counsel as to your own procedure."
"What use is counsel?" said I. "Because when you undertake to do anything by force, no man living can oppose you."
"Well then," he said, "don't oppose me."
"Very well, I shan't," said I.
And thus a dialogue, beginning with reference to one of the opening battles of the major phase of the Peloponnesian War, a dialogue that shows that Charmides and Critias don't know what temperance is, ends with a foreshadowing of the role of Critias and Charmides in the bloodthirsty government of the Thirty Tyrants toward the end of the War.
* The Battle of Potidaea was fought in 432 BC between Athens on one side and Corinth and its allies (including Potidaea on the other side. Potidaea was in an ambiguous position, being a colony of Corinth but also a member of the Delian League dominated by Athens; Corinth and Athens were not on good terms. After tensions became fighting, Athens demanded that Potidaea tear down its walls, expel its Corinthian embassy, and send hostages to Athens. The Athenians, after heavy fighting, want the battles and settled down for a siege. When our dialogue opens, the siege is apparently still going on, because the Athenians are still camped out at Potidaea; it would last two years, draining the Athenian treasury and setting up the conditions for Pericles' fall from power. This apparent indication of context, however, is difficult to reconcile with the fact that the devastating battle mentioned in the dialogue seems to have occurred after the siege was over, on the way back home. That Socrates was a veteran of Potidaea is well-established; his bravery during the battle, in which he saved the life of Alcibiades, seems to have been widely recognized.
* Note that Socrates says that Critias' family -- which is, of course, Plato's family -- has a long history of combining poetry and philosophy.
* Socrates' invented backstory for the charm, in which it is a Thracian remedy, seems to be based on Herodotus (Persian Wars, Book 4.93 and following). Herodotus recounts that the Getae, one of the Thracian tribes, believed that they were actually immortal, and that instead of dying they went to their god Zamolxis.
Since Charmides is one of the perplexed-conclusion dialogues -- it ends with perplexity about what temperance is, rather than any definite positive account -- one should see the main thought of the dialogue to be in what is only hinted. One of the clear major concerns in the dialogue is that temperance or discretion (sophrosyne) must be a practical skill, a craft, making something good. Socrates highlights this toward the end, and it is notable that none of the definitions given by Critias and Charmides make any sense of the idea that temperance is something making people's lives better. Quietness, modesty, minding one's own business, knowledge of one's own knowledge, all fail in some way in this regard. Indeed, they are all quite indeterminate; if temperance were any of these things, an immense number of obviously bad things could be accounted as temperance. And perhaps this is part of the point: Critias and Charmides, future tyrants, only know of a 'temperance' that is compatible with excess in action.
We are in fact provided with several other clues as to what temperance must be: we have the direct connection with beauty, and we have the fact that the discussion is explicitly placed in a medical context. (It's never said, but in context it's difficult to imagine that Charmides' recurring headache could be due to anything but the usual excesses of youth.) We also see that Critias and Charmides have tendencies opposing temperance: Critias is headstrong and unwilling to recognize his limits and Charmides is weak-willed and excessively inclined to obey Critias, while the two of them together are inclined to have their way regardless of what others want. But, as with all perplexed-conclusion dialogues, it is up to the reader to try to solve the riddle.
Quotations from Rosamond Kent Sprague's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 639-663.