The moral dilemma involving Euthyphro and Socrates begins when Socrates challenges Euthyphro by asking him what holiness really is, since Euthyphro claimed he was an "expert" in religious matters. Euthyphro defines holiness as persecuting religious offenders. Socrates found this definition unacceptable, seeing that there are many other more worthy deeds than persecuting people.
Euthyphro went on to say that whatever the God's found agreeable was indeed holy. Socrates then went on to say that the God's fought with each other, so what might be agreeable to one might not be to another. Basically, Socrates was trying to prove that holiness cannot be defined by the Gods. Therefore, neither the Gods nor holiness can be defined because each was inherently based on the other.
This left Euthyphro in a dilemma, unsure of what constituted holiness. He believed that holiness was a kind of justice that was for "looking after the Gods". Socrates again disagreed by saying that the Gods are all powerful and don't need to be looked after in any way. Again, Euthyphro rebutted by saying that sacrifices are made to the Gods in order to grant prayers. All this really accomplished was bringing the argument back the beginning, leaving the dilemma unsolved.