The argument of this dialogue is based largely on "definition by division". Socrates goads Euthyphro to offer one definition after another for the word 'piety'. The hope is to use a clear definition as the basis for Euthyphro to teach Socrates the answer to the question, "What is piety?", ostensibly so that Socrates can use this to defend himself against the charge of impiety.
It is clear that Socrates wants a definition of piety that will be universally true (i.e., a ‘universal’), against which all actions can be measured to determine whether or not they are pious. It is equally clear that to be universal, the definition must express what is essential about the thing defined, and be in terms of genus, species, and its differentiae (this terminology is somewhat later than Socrates, made more famous with Aristotle).
Hence this dialogue is important not just for theology, ethics, and epistemology, but even for metaphysics. Indeed: Plato's approach here has been accused of being too overtly anachronistic, since it is highly unlikely that Socrates himself was such a "master metaphysicist". But the more expository treatment of metaphysics we find in Aristotle has its roots in the Platonic dialogues, especially in the Euthyphro.
The stages of the argument can be summarised as follows:
Euthyphro offers as his first definition of piety what he is doing now, that is, prosecuting his father for manslaughter (5d). Socrates rejects this because it is not a definition; it is only an example or instance of piety. It does not provide the fundamental characteristic which makes pious things pious.
Euthyphro's second definition: piety is what is pleasing to the gods (6e-7a). Socrates applauds this definition because it is expressed in a general form, but criticizes it on the grounds that the gods disagree among themselves as to what is 'pleasing'. This would mean that a particular action, disputed by the gods, would be both pious and impious at the same time — a logically impossible situation. Euthyphro tries to argue against Socrates's criticism by pointing out that not even the gods would disagree amongst themselves that someone who kills without justification should be punished but Socrates argues that disputes would still arise — over just how much justification there actually was, and hence the same action could still be both pious and impious. So yet again, Euthyphro's 'definition' cannot possibly be a definition.
Euthyphro attempts to overcome Socrates's objection by slightly amending his second definition (9e). Thus the third definition reads: What all the gods love is pious, and what they all hate is impious. At this point Socrates introduces the "Euthyphro dilemma" by asking the crucial question: "Is the pious loved by the gods because it is pious? Or is it pious because it is loved by the gods (10a)?" This is where he sets up his typical dialectic. This Socratic technique, essentially an analogy or comparison in this instance, is used to make his question clearer. He gets Euthyphro to agree that we call a carried thing "carried" simply because it is carried, not because it possesses some inherent characteristic or property that we could call "carried". That is, being carried is not an essential characteristic of the thing carried; being carried is a state. Likewise with piety: if it is defined as "what is loved by the gods," it is liked for some reason, not just because it is liked (which would be circular in its reasoning). So the fact that the gods like a pious action does not, de facto, make the action pious. The liking must follow from something, which, in this case, is the recognition that an action is pious prior to its being liked, and not the other way around. Thus the piety comes before the liking both temporally and logically, yet in Euthyphro's definition it is exactly the other way around (it is pious because the gods like it). Therefore Euthyphro's third definition is severely flawed.
To the modern reader, this part of the argument (10a-11a) sounds painfully convoluted. But it had to be written this way, because the Greek of Socrates's time lacked the grammatical terminology to refer to the active voice and passive voice which would have greatly simplified Socrates's expressions. Nor can he refer to Aristotle's Categories, which also goes into great detail on this distinction (treating it as between simple expressions of state and secondary substances). So he explains with detailed examples ('carried', 'loved', 'seen') instead.
Without yet realizing that it makes his definition circular, Euthyphro at this point agrees that the gods like an action because it is pious. Socrates argues that the unanimous approval of the gods is merely an attribute of piety; it is not part of its defining characteristics. It does not define the essence of piety, what piety is in itself; it does not give the idea of piety, so it cannot be a universal definition of 'piety.'
In the second half of the discussion Socrates himself suggests a definition of piety (12d), namely that "piety is a species of the genus 'justice'". But he leads up to this with both observations and questions concerning the difference between species and genus, starting with:
...are you not compelled to think that all that is pious is just?
However, as he then points out a little later, this is still not enough for a definition, since piety belongs to those actions we call just or morally good. However, there are more than simply pious actions that we call just or morally good (12d); for example, bravery, concern for others and so on. What is it, asks Socrates, that makes piety different from all those other actions that we call just? We cannot say something is simply because we believe it to be so. We must find proof.
Euthyphro then suggests that piety is concerned with looking after the gods (13b), but Socrates immediately raises the objection that "looking after", if used in its ordinary sense, which Euthyphro agrees that it is, would imply that when you perform an act of piety you make one of the gods better — a dangerous example of hubris, which gods frowned upon (13c). Euthyphro claims that "caring for" involves service. When questioned by Socrates as to exactly what is the end product of piety, Euthyphro can only fall back on his earlier claim: piety is what is loved by all the gods (14b).
Euthyphro then proposes yet again another definition: Piety, he says, is an art of sacrifice and prayer. He puts forward the notion of piety as a form of knowledge of how to do exchange: giving the gods gifts, and asking favours of them in turn (14e). Socrates presses Euthyphro to state what benefit the gods get from the gifts humans give to them, warning that this "knowledge of exchange" is a species of commerce (14e). Euthyphro objects that the gifts are not that sort of gift at all, but rather "honour, esteem and favor" (15a). In other words, as he admits, piety is intimately bound up with what the gods like. The discussion has come full circle. Euthyphro rushes off to another engagement, and Socrates faces a preliminary hearing on the charge of impiety.