You can read Euthyphro in English at the Perseus Project and in French at Wikisource.
Pretty much everything we know about Euthyphro comes from this dialogue (he is also mentioned in Cratylus, but the mentions don't obviously add anything to what we learn here). He is a mantis, which might variously be translated as a diviner, seer, soothsayer, or prophet.
The archon basilieus, or king archon, was in Socrates' day a yearly position, primarily concerned with organizing religious rites. He oversaw certain serious cases involving impiety. Anyone could bring a charge before the court of the king archon, who would then determine whether the suit were legally proper, and schedule an anakrisis (literally, interrogation) as a preliminary hearing. At the end, formal charges would be drawn up, and a trial (krisis) scheduled. The dialogue opens at the Stoa Basileus, or porch of the king archon, as Socrates is going to the anakrisis for Meletus's charges of impiety and corrupting the youth.
Euthyphro opens the dialogue by expressing surprise at finding Socrates at the stoa of the archon basileus. Socrates explains that he is here because Meletus laid an indictment against him. Euthyphro doesn't know who Meletus is. Socrates says that he thinks Meletus must be wise, since he knows how the youth are corrupted, and he is the only politician beginning in the right way, since he cares for the youth of the city in order to make them good. If he carries on as he has begun, he will greatly benefit the city. Euthyphro replies that it seems to him that Meletus has instead begun by injuring the city, and asks in what way Meletus claims Socrates corrupts the youth. Socrates replies that he claims that Socrates makes new gods and does not worship the old ones.
Euthyphro responds that it is because of Socrates' daimonion, and sympathizes since he has had people laugh at him for talking about divine things and foretelling the future in the Assembly, saying that one must fight them closely. Socrates says that if it were only laughter, this would be insignificant; but he fears that they are in earnest. Euthyphro says that perhaps Socrates will be as successful as Euthyphro thinks he will in his own. This gets them talking about Euthyphro's case, in which he is prosecuting his father for impiety, and Socrates is surprised, since prosecuting one's own father would ordinarily be considered impious. They then discuss the nature of the pious (hosios) and the impious (anosios). Euthyphro's proposals turn out to have problems, so that they end up circling around, and Socrates says that they must keep going until it's sorted out. But Euthyphro says he has to go, and Socrates ends with the comment:
What a thing to do, my friend! By going you have cast me down from a great hope I had, that I would learn from you the nature of the pious and the impious and so escape Meletus' indictment by showing him that I had acquired wisdom in divine matters from Euthyphro, and my ignorance would no longer cause me to be careless and inventive about such things, and that I would be better for the rest of my life. (15e-16a)
General Comments on the Plot
It is worth noting that while Euthyphro is prosecuting, and Socrates defending, impiety charges, the charges in each case would have been very different. Euthyphro's charge against his father is an oral charge, for private crimes, and (as Socrates notes to him) usually only allowed for relatives of the victim; the charge Socrates is facing is a written charge, suggesting that the offense is against the city itself, and as such could be brought by any citizen. Murder was a religious crime; a fact often forgotten when people discuss what the dialogue says about impiety.
Euthyphro's case is remarkably complicated. His family had a farm at Naxos. The island of Naxos was conquered territory (it had been crushed by Athens for trying to leave the Delian League), and one way in which the Athenians handled conquered territory was to portion out land grants from the territory to Athenian citizens to colonize it (in exchange, the territory's tribute to Athens was reduced); this made the territory what was called a 'cleruchy'. The cleruchy at Naxos ended in 404, at least five years earlier, so the crime Euthyphro is prosecuting is already five years old. A drunken laborer killed a house slave; Euthyphro's father bound the murderer and put him in a ditch, then sent to Athens for information on how to handle the situation. However, the laborer in the ditch, not given any amenities, died before any message came back from Athens. Euthyphro is prosecuting his father for murdering the laborer. One puzzle presents itself immediately: the advice Euthyphro's father needed was religious advice; so why didn't he get it from Euthyphro? He might not have been the right kind of authority -- Euthyphro's father sent for the advice of an exegete, which as far as we can tell would have been someone specializing in religious law, and there's no indication that Euthyphro had that kind of expertise. The fact that Euthyphro must have spent a good part of his life at Naxos, however, explains why he can both be an Athenian citizen (he has spoken in the Athenian assembly) and why Socrates always speaks of 'the Athenians' as if Euthyphro weren't one of them.
Euthyphro clearly knows Socrates well; he recognizes Socrates on sight, knows that he spends much of his time at the Lyceum, and knows that Socrates claims to have a divine sign, his daemon or daimonion. Euthyphro also regards this divine sign as indicating a common ground with Socrates: in his mind, Socrates and he are alike. This is not unreasonable, nor does this appear to be (as some commentators seem to suggest) a bit of irony: both Plato (Apology 40a, Phaedrus 242b-c) and Xenophon (Memorabilia 1.1.3) treat Socrates' divine sign as equivalent to divination or the work of a mantis, although it is mantic in a peculiar way (it is worth also remembering that Plato depicts Socrates as having premonitory dreams and both Plato and Xenophon take Socrates to be supported by the oracle at Delphi). When we are introduced to Euthyphro as a mantis, it is absolutely essential to not to read back modern ideas of quackery or superstition into this; ancient Greeks often took omens very seriously in political and military matters (cp. Nicias in Laches), and manteis had a generally respected role in their society, although there is certainly evidence that not everyone gave them the same credit (e.g., Laches in Laches). Why then do people laugh at Euthyphro? Perhaps a comment by Thucydides (History 8.1.1) might indicate it; he notes that after the Syracusan disaster, the Athenians became angry at interpreters of oracles and manteis, whom they blamed for misleading them as to the viability of the expedition. The intervening fifteen years were not especially shining for Athens, and there's some reason to think that the reputation of diviners recovered only very slowly after that (although it did recover).
I find that Euthyphro is often treated as if he were either stern and close-minded or fanatical; but there is no indication at all in the dialogue of either. He is very sympathetic to Socrates -- perhaps this is partly politeness, but there is no question that he expresses clear sympathy and support for Socrates. He even seems to joke around with Socrates at one point (when talking about Daedelus). He does hurry away at the end, but we do not actually know the reason why. It could be, as usually assumed, that he was annoyed or angry; but since he quite clearly hadn't been originally planning to have an extended philosophical discussion on the nature of piety as he went about his business, it could equally well be anything else. (It's worth pointing out as well something that occasionally gets noted by commentators: given that Euthyphro's motive for the suit seems to be to purify himself from ritual pollution, we can't even be sure that Euthyphro expected or even wanted his father to be convicted, since from his perspective simply bringing the matter to the notice of the proper court may be enough.)
Part of the significance of the discussion is that Euthyphro is actually someone whose 'career', as we would call it, concerns divine things, and who has a reputation for devoting himself to these things, even if he does get laughed at for it; if anyone has a background that promises insight into what piety is, he certainly does. Meletus, who is only an obscure poet whose name Euthyphro doesn't even recognize (and therefore who does not have a reputation for his insight into divine matters), is certainly not better placed than Euthyphro to understand piety and impiety. (Contrast this with Socrates; Euthyphro recognizes him personally, is astounded to find him on the porch of the king archon involved in an impiety suit, and immediately thinks the charges against him are absurd.) The dialogue is often treated as if it were intended to put the concept of piety itself into question, but the primary idea put into doubt is not piety but rather impersonal justice; the discussion of piety at least partly makes a point about how human beings can or should approach matters of justice. And what Euthyphro and Meletus both share is the attempt to go around charging people with impiety for abstract and general crimes rather than things that have been done to them personally. Piety itself survives unscathed, if not yet defined; it's not so clear that impersonal justice does.
The discussion of piety itself goes through several stages. Euthyphro begins by saying that the pious is to prosecute the wrongdoer, and for his own case he gives the divine precedent: Zeus bound Chronos, and Chronos castrated Ouranos, for unjust actions. Socrates, suggesting that his disinclination to believe such stories is a reason for his indictment, asks if Euthyphro really believes these stories of war among the gods, and Euthyphro replies that he does. (This links, I think, Euthyphro to Theaetetus and to Cratylus, and also sets up the obvious problem for the next suggested definition.)
Socrates remarks that Euthyphro has not actually given the idea/form (eidos) that everything pious shares. So Euthyphro tries again: "what is dear [prosphiles] to the gods is pious, what is not is impious"(7a). This idea of the pious as being what is lovely to the gods founders, however, on the fact that he just said that the gods disagreed.
This leads to the next idea, that "the pious is what all the gods love, and the opposite, what all the gods hate, is the impious" (9e). This introduces the most famous passage in the dialogue, what is often known as the Euthyphro Dilemma: Is the pious lovely to the godsbecause it is pious, or is it pious because it is lovely to the gods? It is important to grasp, however, that Socrates does not at any point argue that the pious is not what is lovely to all the gods; what he argues is that Euthyphro in saying this has not told him what the being (ousios) of the pious is, but only an adventitious feature (pathos) or something that happens to it. So they are back at the beginning; and Euthyphro remarks that he can't tell Socrates what he thinks about the pious and the impious because any statement they put forward moves around rather than sitting still, and that it is Socrates who is doing it.
At this point, Socrates suggests a new starting point: "all that is pious is of necessity just" (11e), and from this they conclude that justice is the genus and piety the species (or part). (It is perhaps worth noting that this approach is reminiscent of the approach we'll see used more extensively in the Sophist and the Statesman.) This raises the question of which part it is, and Euthyphro suggests the next possibility: "the godly [eusebes] and pious is the part of the just that is concerned with the care [therapeia] of the gods, while that concerned with the care of men is the remaining part of justice" (12e). This obviously raises the question of what care is; in most cases it means to benefit and make better, but for a human being to try to make the gods better would obviously be impious. So in this case, care must be service/subordination [hyperetike] to the gods.
Socrates then asks the obvious question: what is the achievement at which service to the gods aims? Or in other words, what do the gods achieve through our service? Euthyphro replies that it is many fine things (polla kai kala), but, of course Socrates is not going to be put off by this, so Euthyphro says the pious is to say and do what pleases the gods in prayer (euchomenos) and sacrifice (thuos). Socrates points out that this means service is to give gifts and to beg, so it seems that piety is a kind of trade. But this runs right back to the problem: we seem to be saying that the pious is to benefit the gods by giving them something that they need and that therefore will make them better. Euthyphro, however, denies that it is a matter of benefiting the gods; what the gods receive are reverence (time), honor (gera), and gratitude/graciousness (charis). But, Socrates points out, this then means that the pious is what is lovely to the gods, and we have gone in a circle again.
And that is where it comes to an end. We perhaps have more than we might think. It is notable that the claim that the pious is the part of justice concerning the gods is never refuted (it is also a claim that Plato's Socrates makes elsewhere); Euthyphro's attempt in this direction fell apart solely through his understanding of 'care of the gods'. And recall the beginning -- Socrates praising Meletus, albeit ironically, for seeking to make young and old better. And recall exactly how it ends: Socrates says that if Euthyphro had taught him what piety is, he would live a better life for the rest of his life. I don't think these are just trimming. Plato generally doesn't do mere trimming; he does not just argue with his explicit statements but also with his insinuations. But even setting that general consideration aside, it seems unlikely to be an accident that Socrates emphasizes this idea at both the beginning and the end of the dialogue.
Quotations are from G. M. A. Grube's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 1-16.