The Structure of the Euthyphro is as follows:

After an introductory conversation in which we find out why Euthyphro and Socrates are hanging around the Stoa of the King, Socrates suggests that Euthyphro should teach him what piety is. Accordingly, he asks Euthyphro to explain to him what one thing makes all the pious things pious. This is a "What is F?" question: "What is piety?" Plato's Socrates' favorite sort of question is the "What is F?" and it is the subject of much rich thought, for it is the first sustained attempt in the tradition Plato sits at the start of to identify a definition, a universal, an essence, or whatever we think it is that Socrates wants.
  1. Euthyphro's First Response: 5d-6e
  2. Euthyphro's Second Response: 7a-9d
  3. Emended Second Response: 9d-11e
  4. Socrates' suggested procedure: 11e-14c (where did E go wrong? is there a right answer there?)
  5. Euthyphro's Third Attempt: 14c-15b

Before going through the dialogue, look at its conclusion (16a). Socrates there claims very clearly that he does not know what piety is. This is one of Socrates' famed "disavowals" of knowledge. Socrates claims, namely, right at the end of the dialogue not to know what virtue is. Dialogues that end in puzzlement are called aporetic after the Greek word for 'puzzle' or 'quandary.' The Greek word is also an English word, aporia.

Euthyphro's First response (piety is prosecuting a wrongdoer) identified too small a part of piety. In other words, the answer had a scope problem: it's scope was insufficient to answer the question (6d).
What Socrates wanted was one form that all piety shares, and that nonpious things do not have. He wanted a criterion by which to identify all pious things and only pious things (6d). The scope of such an answer would fit the scope of piety perfectly.

Euthyphro's Second Response (the pious is what god loves, 7a) was Formally correct: it had the right scope and was universally applicable to piety, but it was incoherent, because different gods love (and hate) different things (7b). Thus something could be loved by one god and hated by another, which would make it pious and not pious (8a-b). That is only incoherent if we assume that piety is not something that consists in the relationship between a god and a thing or if we assume that it is true that some gods love different things. In other words, it assumes that piety is not like taste: there is nothing incoherent about saying that something tastes both good and bad (OJ for instance tastes good to most people, but not after brushing your teeth, and it always tastes bad to some people). It also assumes there are gods, which is problematic for some, whether because they are atheists or monotheists or think the gods don't care about such things (or any number of other ways that god(s) might not fit the claim "god loves the pious").

Euthyphro's Emended response: what all the gods love is pious (9d). Problem: piety exists independently of whether gods love something or not (10a, 10e). The causal direction is as follows: a thing's piousness causes it to be loved by the gods. BUT the gods' love does NOT cause things to be pious. That seems like a sensible claim, but consider the Old Testament: why are the 10 commandments right? on a very plausible interpretation of the OT, they are right because god said so. There is no big worry about why god says so: god says so, and so, because of that, they are to be obeyed. This has huge implications for what sort of thing god(s) is.

That brings us to an important distinction:
Accident vs. essence (11a)
The essence of humanity is rationality and animality (at least on one going definition of humanity). It is accidental that some humans are dark or light-haired, that they weigh 100 or 200 pounds, that they know Tagalog, Malagasy, English, or Latin. The dialogue suggests that Euthyphro identified an accident of piety (that gods love it) rather than its essence. Socrates wanted its essence.Importantly, this is a huge point for Platonic Forms and also for the history of philosophy: this is the first theorization of something like essence vs. accident, a powerful conceptual analysis.

Socrates' attempt to show Euthyphro how to answer (11e and following): The discussion became circular after Socrates got Euthyphro to agree that piety is a part of justice. But wait, an important observation is that Socrates thinks that Piety is a part of justice: that seems to be an important moral claim which Socrates believes. Does he know it? He says he does not know what piety is at the end of the dialogue, so this claim is in tension with that claim.
Socrates agrees that piety is care of the gods at 12e: note well that this is another claim that seems to be knowledge about piety, but he does not know what sort of care one could offer to a perfect thing that lacks nothing and needs nothing (i.e. to a god). Care usually improves things, but the god(s) cannot be improved.
It is important to note that this part of the dialogue is meta: it talks about what is happening in the dialogue itself, what moves the characters are making, and suggests ways to improve them. Plato often goes meta. The toolchest of philosophy is being created right before our eyes here.

At 14b, Euthyphro suggests another answer: conventional religious actions such as sacrifice, prayer, etc. are care of the gods, but their opposites (whatever the opposite of prayer and sacrifice is: blasphemy and sacrilege?) are impious. Socrates replies that Euthyphro had been on the verge of giving him the kind of answer he wanted at that point! What would that answer have been? And, importantly, is this a clue left by Plato's Socrates that we can use to figure out an answer? One suggestion is that we care for the gods by caring for ourselves and increasing our own virtue.

The examination of Euthyphro's third answer reveals that piety is a kind of equal exchange between humans and gods: we offer sacrifice, honor, reverence, and prayer, and they do good things for us. This definition not only reverts to the previous "the pious is what the gods love" answer (15b), but also involves a sort of insult to gods. The insult is that they might need something from us and so not be perfect. It is a do ut des model of human-divine interaction, a transactional model, and as such is a non-starter for thoughtful deists.

Socrates believes that the sort of things he asks about (what is piety, what is courage, what is virtue, etc.) are knowable and explainable by humans. Thus he must think that there is some chance that Euthyphro knows and can explain piety. That chance is increased by the fact that Euthyphro is prosecuting his father, because prosecuting one's father is, prima facie, an impious act according to Athenian beliefs.

Socrates thinks along the following lines: someone who claims to be pious and yet prosecutes his or her father MUST know what piety is. Thus action implies knowledge for Socrates. Conversely, we will see that knowledge implies action (if you know what is right, you cannot not do it). Note that these two claims are not made directly and explicitly in the dialogue: they are 'implicit' epistemological claims about virtue.

Given that Socrates thinks his questions are answerable by humans, he is not disingenuous when he approaches people and takes them at their word if they claim to know about piety, courage, etc. He may well suspect that they are mistaken about their state of knowledge, but he must believe that there is a chance that they know. Furthermore, if he is to have a chance at finding out what he wants to know, what else should he do? it would be counterproductive and unpleasant to challenge people by saying that they don't know what piety or courage is right from the start. It is much more pleasant to ask them to teach him. It is pleasant and also that way they are more likely to trust him and to be willing to talk with him.

Suggestion about the implicit assumptions behind what Socrates is doing: Socrates makes some simple plausible assumptions: