After running into Euthyphro outside of king-archon's court and hearing about why Euthyphro is there, Socrates is not convinced that Euthyphro prosecuting his father for murder is the just or pious thing to do. He asks Euthyphro to teach him about what piety and impiety are, so that he can see for himself whether what Euthyphro is doing to his father is a pious act. This, then, begins the heart of the dialogue--a rigorous discussion about what piety and impiety are.

(1) Euthyphro's 1st Attempt
Euthyphro first tries to explain to Socrates what piety and impiety are by giving him examples. He says, "the pious is to do what I am doing now, to prosecute the wrongdoer, be it about murder or temple robbery or anything else, whether the wrongdoer is your father ot your mother or anyone else." (5e)

•  Socrates' Objection:  Socrates complains that he did not ask for a list of the pious and impious things; he wanted to know what piety and impiety are.
  

To see why he was frustrated, consider an analogous case: imagine that you and a friend are sharing a pitcher at some bar and you are considering home brewing your own beer. The only problem is that you know hardly anything about beer. So you ask your friend, who professes to be rather knowledgeable about such matters, "what is beer?" He then answers as follows: "Well, this stuff that we're drinking is beer, and the stuff that the dude over there is drinking is beer, and the stuff we drank last night is beer, etc." Understandably, you would be pretty annoyed, because what you wanted was an explanation of what beer was--i.e., what it was composed of, what ingredients are essential to beer and make it different from, say, wine or kool-aid. By simply pointing out instances of beer is of very little help to you. Likewise, Socrates is interested in what piety is--i.e., what it is "composed" of, what things are essential to it and make it different from, say, justice or love.

(2) Euthyphro's 2nd Attempt
Euthyphro then defines piety and impiety as follows: "what is dear to the gods is pious, what is not is impious." (7a).

• Socrates' Objection: Earlier in the dialogue (6c) Socrates has confirmed that Euthyphro believes in the greeks gods and all of the stories about them--e.g., he believes that they fight, and that there is war between them, and that they disagree about many things. Recalling this, Socrates points out that this will prove problematic for Euthyphro's definition of piety. For if what is dear to the gods is pious (and what is not dear to the gods is impious), and yet if the gods disagree and fight about what is dear to them, then it will turn out that one and the same action will be both pious and impious (since it will be dear to some gods and not dear to others).

(3) Euthyphro's 3rd Attempt
After some prompting by Socrates, Euthyphro next settles of the following definition of piety: "the pious is what all the gods love, and the opposite, what all the gods hate, is impious." (9e)

• Socrates' Objection:  It is here where Socrates brings up (what we called in class) the Euthyphro Problem. He asks of Euthyphro whether "the pious is loved by the gods because it is pious, or is something pious because it is loved by the gods?" (10a)

(4) Euthyphro's  4th Attempt
Again prompted by Socrates, Euthyphro next tries to say how just actions and pious actions are related. He then claims that "the godly and pious is the part of the just that is concerned with the care of the gods, while that concerned with the care of the men is the remaining part of justice." (12e)

The idea is something like this: justice covers a lot of things--things having to do with gods and men. Piety, on the other hand, only has to do with the just things that concern only gods (and not men). So while all pious things are just, not all just things are pious.

• Socrates' Objection: Socrates first concentrates on what exactly Euthyphro means when he says that "the pious is the part of the just that is concerned with the care of the gods." For he thinks that "care of" indicates bettering something, as it does when someone cares for dogs or horses or cattle. But surely, Socrates argues, the gods aren't bettered by our pious actions, since nothing we can do can improve the gods.
  

(5) Euthyphro's 5th Attempt
Euthyphro then amends his account by claiming that the kind of care he was talking about was the kind of care "that slaves take of their masters." (13d) And goes on to endorse the view that piety is (as Socrates puts it) "a sort of trading skill between gods and men." (14e)

• Socrates' Objection:  Socrates challenges Euthyphro by claiming that gifts are beneficial to the receiver, but, he asks, how could the gods possibly benefit from anything that we do? 

In response, Euthyphro then claims that our serving the gods simply pleases them, or is dear to them. At which point, Socrates counters that that Euthyphro has now come full circle, since it looks like he is back to the claim that what is pious is that which is pleasing or dear to the gods. Socrates then starts in again, saying that he still wants to know what piety is, and that they "must investigate again from the beginning what piety is." (15d).  Frustrated, Euthyphro leaves. And so the dialogue ends.

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Sources:

Plato, Euthyphro in The Trial and Death of Socrates, translated by G. M. A. Grube, Hackett Publishing (1975).

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