When Socrates and Euthyphro meet, Socrates clarifies for Euthyphro the charges that the state has brought against him and Euthyphro is disturbed to hear about the trouble of his friend. He says that he too has been involved in a rather unpleasant set of charges, namely his own accusation against his father.
Socrates is quite surprised to hear this because in ancient Greece it was considered very bold to officially accuse one's own family member of anything, and mortals who did such were not looked upon kindly by the Greek Gods. Euthyphro admits that he is prosecuting his father for the murder of a servant and consequently, he is considered by his fellow citizens and statesman to be acting "impiously". Euthyphro, rather arrogantly, asserts that the people know not what impiety truly is, for if they did they would not consider his actions to be of the sort. This assertion indirectly indicates to Socrates that Euthyphro has knowledge of piety and impiety, and Socrates draws and analogy between his own case and the case of Euthyphro. If Euthyphro can explain to Socrates the meaning of impiety, perhaps Socrates can argue better against his own charges and so he asks Euthyphro to kindly teach him about piety, thus assuming the role not of the teacher, but of the student.
"Tell me then, what is the pious, and what the impious, do you say?", says Socrates. And the first answer Euthyphro gives is that it is to prosecute the wrongdoer, as he is doing in the case of his father. Socrates rightly points out that Euthyphro gave him an example rather than a definition and asks him to clarify his answer in the form of a definition. This bit of dialogue is important because it introduces the idea behind much of Plato's writing insofar as there is an instance here of the 'one over many principle' which constitutes the theory of Plato's Forms. Again, Euthyphro has given and example of piety (of which there are many other examples) when instead Socrates wants to know what piety actually is (of which there can be only one definition). Socrates wants a specific account of piety-a uniqueness requirement for the Euthyphro's answer.
The second answer then given by Euthyphro is "that which is dear to the gods is pious, what is not is impious". And with that Socrates is more pleased since he has now received an answer closer to his expectations of what the form of the answer should be, namely, a definition or account of piety. The problem with this answer as Socrates argues, is that the gods love too much. They love and hate different things and between themselves they disagree on what is to be loved and what is to be hated. So if piety is that which is loved by the gods, according to Euthyphro, one cannot tell what is more pious or not since there is disagreement among the gods about what they love. Socrates is looking for that form or idea itself that makes all things pious; a causal or explanatory requirement to add to the uniqueness requirement for the answer.
The third suggestion made by Euthyphro is simply a modification of his second answer: that which all of the gods love. Socrates suggests that they examine the soundness of this definition before accepting it, as is the case with most dialogues of Plato. An answer is scrutinized by Socrates (and other Platonic characters) to ensure that it is sound in all cases before accepting it as valid and true. Unfortunately, in the course of their examination Socrates and Euthyphro discover that the given answer is explained by piety, instead of it itself explaining piety as a proper definition should. To clarify, if the god-loved (E's definition of piety) and piety (the word we are looking to define) are the same, then the pious would be loved because it was pious, just as the god-loved would be loved because it was god-loved. One perspective is that the god-loved was loved by the gods because it was loved by the gods (as we already said god-loved was to mean). Another perspective is that the pious was pious because it was being loved by the gods. And the problem with this is that one definition has two different meanings-one is that if "X" is being loved by the gods, then "X" is pious; the other is that if "X" is to be loved by the gods, then "X" is pious, and these two interpretations are different when they should be the same. Socrates requires that the definition gives the "essence", or true nature of piety, in one clear definition.
At this point, Euthyphro is rather frustrated since every definition he gives is picked apart by Socrates and discredited. This is typical of Plato's writing because it not only exalts his admiration of Socrates' argumentative skill, but also illustrates the nature of philosophical discussions. They are discussions filled with arguments, usually one from Socrates and one from another character, and they wrestle back and forth with points and problems raised in each other's arguments until an answer can be reached, though often it is the case that no conclusive answer can be reached.
The rest of the dialogue is not quite as substantial as the earlier parts, but we are given the fourth and final attempt to define piety from Socrates himself. His answer is that piety is the part of justice concerned with care of the gods. He questions the relation between piety and justice and wonders if one is a part of another. Euthyphro is unable to contribute much more to the conversation because of his frustration and apparent lack of skill in arguing. He makes excuses for himself and ultimately avoids the questions of Socrates by excusing himself to take care of other, more important matters. The fact that Euthyphro does not enjoy doing the dialectic is indicative of most other people during the time of Socrates and Plato. Philosophy was held in high regard by philosophers, and philosophers only.