Cousin rightly notes that there is a great deal in the dialogue that is comic. Euthydemus is also the apparent origin of one of the most famous sophisms of all time (298d-e):
You will admit all this in a moment, Ctesippus, if you answer my questions, said Dionysodorus. Tell me, have you got a dog?
Yes, and brute of a one, too, said Ctesippus.
And has he got puppies?
Yes indeed, and they are just like him.
And so the dog is their father?
Yes, I saw him mounting the bitch myself, he said.
Well then: isn't the dog yours?
Certainly, he said.
Then since he is a father and is yours, the dog turns out to be your father, and you are the brother of puppies, aren't you?
You can read Euthydemus online in English at the Perseus Project, and in French at Wikisource.
(in order of appearance)
Crito is a wealthy friend of Socrates; he is mentioned in the Apology, has a dialogue named after him, and is also a character in Phaedo. He is also found in Xenophon's Memorabilia and Symposium. He is notable for being the man who was willing to pay for all of Socrates' expenses at and after his trial. He has a son, Critobulus, who is not in the dialogue but plays a role nonetheless, since Crito is trying to figure out the best way to have him educated. Critobulus will become one of Socrates' close students. Despite the connection with Socrates, however, Crito seems not to have been particularly invested in philosophical inquiry himself, and often seems to have a benign but slightly skeptical attitude toward it.
Euthydemus is a famous Sophist, who became a Sophist after some time teaching how to fight in armor. As we will see in the dialogue, he handles argument almost as if it were fighting in armor. Most of our information about him comes from this dialogue, although Plato mentions him briefly in Cratylus (386d). Aristotle also mentions him briefly in Rhetoric (1401a) and Sophistical Refutations (177b). He is to be distinguished from another Euthydemus who is mentioned by Xenophon as Socrates' student.
Euthydemus' brother. He is also mentioned by Xenophon in the Memorabilia (3.1.1), who notes that he taught generalship.
Clinias son of Axiochus
He is the cousin of Alcibiades.
Ctesippus of Paeania
Ctesippus is the same as the Ctesippus in Lysis. (Notice the curious fact that he is again distinguished not by the name of his father but by the deme in which he is registered as a citizen of Athens.) He is outspoken here as there, and I think in this dialogue he ends up being one of Plato's most vivid characters -- a roguish young man, a little wild, with a tremendous laugh, a sarcastic sense of humor, and a refusal to back down.
There is also a crowd of other people, mostly boys.
The Plot and The Thought
Euthydemus has an interestingly complex structure, with a complex narrated dialogue that is related within the framework of another dramatic dialogue that is fairly substantive in its own right; Crito will break into the narrative dialogue to comment on the behavior of the Sophists.
Crito opens our dialogue by asking Socrates who he was talking with in the Lyceum yesterday. Socrates replies that it was with Euthydemus and Dionysodorus, two brothers originally from Chios, although more recently hailing from Italy. Socrates says that they are marvelously wise and that he wants to become their student, and that Crito should perhaps bring his sons along, too; this is apparently ironic, but, if so, Crito misses it and asks Socrates to tell him something about their wisdom. So Socrates begins to narrate the dialogue with the two brothers.
In the narrated dialogue Socrates says he was in the undressing room and about to live when his divine sign forbade him to go. So he waited, and in came Euthydemus and Dionysodorus, and shortly afterward Clinias and Ctesippus. Clinias saw Socrates and came to sit by him; Euthydemus and Dionysodorus soon afterward sat by them. Socrates remarked to Clinias that the two brothers taught fighting, but they quickly say that they have moved on to other things -- teaching virtue, in particular. Obviously, this is the sort of thing that interests Socrates, and he asks them for a demonstration.
As Euthydemus begins to question Clinias, Dionysodorus remarks to Socrates that no matter what Clinias says, Euthydemus will refute him. This is in fact true; in an extended back and forth that Plato describes in vivid athletic terms, the two refute Clinias coming and going. Socrates jumps in with an attempt to spare Clinias yet another refutation, remarking that Clinias should understand that the two are engaging in "dancing and sport" as a part of the "sophistic mysteries" and are engaging in this "frivolous part of study" like people pulling chairs out from under people who are about to sit down as a practical joke (277d-e, 278b). He urges the two sophists to move on to the serious hortatory work of teaching Clinias virtue. To clarify, he gives a demonstration himself, leading Clinias to see that wisdom is the secret to good fortune, and that it makes any good a greater good.
The sophists respond by arguing with Socrates and Ctesippus (whom they rile a bit), and after this continues a while, Socrates tries again to show the way to engage in hortatory argument toward virtue by picking up with Clinias where he had left off and discussing the art that gives happiness and wisdom. Crito (notably) breaks into Socrates' description of this dialogue, very impressed by Clinias's answers, saying that if he was saying these things he doesn't need to be educated by Euthydemus. Socrates says that perhaps he is misremembering and it was really Ctesippus; to which Crito responds that it doesn't sound like Ctesippus. Socrates takes this interest on the part of Crito to begin a dialogue with Crito on the same subject. We then return to Euthydemus, who argues that anyone who knows anything knows everyone. In this round of argument, Socrates manages to rile Euthydemus, and at one point manages to get the two brothers tangled up in each other's arguments. Ctesippus, in the meantime, is starting to get the hang of the sophists' arguments, and manages to score a point or two against them, although they continue, slippery to the end.
Everyone applauds the brothers' skill at argument, and Socrates ironically advises them that, since they were able to teach Ctesippus so quickly, they should perhaps only argue with each other and their paying students, lest, in arguing with a great many, everyone might learn their art and they would lose money. Socrates finishes up his narration to Crito by telling him to find a way to get in on the classes of the two brothers so that they can learn wisdom. Crito, however, is utterly unimpressed (as Socrates no doubt intended), and describes how he had met up with someone who had been in the crowd, who remarks that the display shows that philosophy is worthless. Crito disagrees with that, but reproaches Socrates for arguing with such people. Socrates has some rather biting comments about this kind of person, remarking that such people think themselves wiser than anyone else, but that their views are more plausible than true.
Crito remarks that he is still uncertain about what to do with his sons; Critobulus is getting to the point where he needs education; whenever he is in Socrates' presence, he begins thinking that his education is the most important thing to look to. But the people who purport to educate men are just absurd, so that he doesn't know how to encourage the boy to take up philosophy.
Socrates' response is to point out that most practitioners of almost any art are rather mediocre at best, and that this is not a reason to avoid them. And thus he ends the dialogue with what I think should be seen as the entire point all along, the idea that philosophy itself is not a matter of who is teaching you but of what you, yourself, are doing:
Then don't do what you ought not to, Crito, but pay no attention to the practitioners of philosophy, whether good or bad. Rather, give serious consideration to the thing itself: if it seems to you negligible, then turn everyone from it, not just your sons. But if it seems to you to be what I think it is, then take heart, pursue it, practice it, both you and yours, as the proverb says. (307b-c)
* Connus, Socrates' music teacher, is attested elsewhere. Plato mentions him again in Menexenus. There was a comedy entered in the Dionysia of 423 BC by Ameipsias named after him, in which Socrates was one of the characters (the play itself has not survived except in a few fragments). It has the memorable description of Socrates as "best among few men and most absurd among many". Interestingly, Connus placed second in the contest, beating out the original version of Aristophanes' The Clouds, which is about Socrates himself. Aristophanes seems not to have liked placing third; he reworked the play significantly, and this is the version of The Clouds we currently have, since the comedy as we have it makes explicit reference to the prior failure of the play. There is also a play, also not extant, by Phrynichus called Connus. Connus was famous for having been a great musician who failed and ended up trying to eke out a living by teaching others music; no doubt that would be why he accepted the old man Socrates as one of his students.
* If we are allowed to use fancy Greek words, Euthydemus is a contest between eristic and elenchus. Eristic is related to the word eris, meaning 'discord' or 'strife', and was a pugnacious question-and-answer method to train students how to refute arguments, any arguments. Socrates's dialectic, of course, is also a question-and-answer approach to education, and it is clearly one of the purposes of the dialogue to show people who confused the two how very different they were in practice. Thus we have repeated contrasts between the eristic of the sophists and Socrates's discussion with Clinias.
Isocrates attacks eristic in his speech Against the Sophists.
* Rosamond Kent Sprague has an excellent little discussion of Euthydemus as a philosophical examination of education: Two Kinds of Paideia in Plato's Euthydemus. A good comment on the contrast:
Another means of contrast adopted by Plato is to show, in the case of eristic, that its practitioners care nothing for the future of their pupils; all they require is someone who is willing to answer 275C. Their purpose is to give a demonstration of eristic tricks with a view to attracting students and making money. Socrates and his friends, on the other hand, desire that the young man "become as good as possible" 275A. Socrates' treatment of Cleinias is kindly, leisured, and adapted to the needs and abilities of the particular pupil before him; the sophists, on the other hand, indulge in a rapid-fire procedure that reaches no definite conclusion and that takes no account of individual differences. Plato emphasizes, further, the ease with which the eristic technique can be learned, (we in fact see Cleinias' admirer, Ctesippus, acquiring the knack before our very eyes 299Eff.), and surely implies, if he does not say, that one hardly becomes a master of dialectic over-night. (In the Republic, as we know, he lays down the lengthy course of studies required in detail.)
Quotations are from Rosamund Kent Sprague's translation in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 708-745.