The "Euthyphro dilemma" (also known as the "Euthyphro problem") is the term used by philosophers and theologians to refer to the following problem believers in a traditional Judeo-Christian God are presented with:
Did God decide what goodness is? If so, then 'good' is more or less the arbitrary decision of a frightening being to which we cannot relate, and that being could just as easily have made murder and stealing the ultimate moral actions without any contradictions. On the other hand, if God did not decide what goodness is, he cannot truly be omnipotent.
This problem is called the Euthyphro dilemma because of its original formulation as set down in Plato's dialogue, The Euthyphro. In it, the character of Socrates says to Euthyphro, a self-declared prophet and pious man:
"Consider this: Is the pious being loved by the gods because it is pious, or is it pious because it is being loved by the gods?" (Euthyphro 10a)
In the times of ancient Greece, this did not present readers with the kind of paradox it has provided modern believers. That is mainly because the Greeks did not believe in the omnipotence of the gods, and with good reason - after all, their gods were clearly anthropomorphic and tended to hurt, punish and even castrate each other; none of which are terribly easy to do to omnipotent beings. The question is almost rhetorical for Euthyphro, for obvious reasons.
In this dialogue, Euthyphro clearly chooses the first option, which in our modern formulation would correspond to "God (or the gods) simply implements goodness, he does not choose what it is." This is also the view taken by thinkers like the Saadia Gaon. The "God arbitrarily (or unknowably) decides what is good" view is taken by René Descartes and others.
There are also views that avoid making a choice. Thomas Aquinas defines the doctrine of divine simplicity, which simply put states that God is goodness, so he cannot be spoken of as "deciding" what goodness is or as "coming after" goodness. The nominalist viewpoint was that universals such as "goodness" do not exist, and God does not have properties, rendering the dilemma meaningless.
In recent times, even more interesting stances have been taken, such as the view that God is a utilitarian, and to say "God is good" simply means "God wants what maximizes happiness for humanity" (or "for all creatures with feelings," if you want to make Peter Singer happy). Of course, whether any of the so-called "solutions" to the Euthyphro dilemma are compatible with the Judeo-Christian biblical God that Descartes, Saadia Gaon, and the Nominalists were trying to help out of this messy theological morass is highly questionable.