Many Christians seem to think that atheists bring up the topic of the Euthyphro dilemma in order to "disprove" god (we cannot disprove what was never proved in the first place). It's not about disproving god, it's about refuting the basis for the moral arguments made by all theists (not just your religion).
While the original Socratic dialog discussed the relationship between morality and religion, and the dialog itself discusses the relationship between the Greek religion and morality, many believe Plato was disproving the whole concept of objective morality, not just whether or not the Greek gods could account for objective morality. He probably framed it in terms of the authority of the Greek gods because that was how most people of his civilization thought about morality.
I'm going to go less broad than that for now. The dilemma applies to any authority that is claimed to be the source of morality. It doesn't matter if the authority is real or fake, a god or an ethics professor. It doesn't matter what properties the authority has. The Euthyphro dilemma shows that we cannot derive morality from authority.
Dilemma Rephrased
Allow me to rephrase the dilemma a bit in order to adapt it to the way modern people talk about the dilemma. If someone claims that morality comes from a given authority, how do we know that the resulting moral claims are in fact moral? How do we know if the authority is good? How do we know if the commands of the authority are good?
The only way to know that the commands of the authority are morally good is to develop a definition of morality that is independent of the authority, but if we do that, then the definition of morality is the source of morality, not the authority. If we do not do that, then we have no way to objectively determine that the commands of the authority are in fact good. In the latter case, morality is simply not objective, but the whim of whoever is claiming to speak for the authority.
False Dichotomy Objection
The standard response is to claim that the dilemma is a false dichotomy fallacy because there is a third option that was not considered in the original dilemma. If we define the authority as having the property of being good, then we know that the commands of the authority are good.
This objection is quite obviously circular logic (a begging the question fallacy). You cannot smuggle the conclusion into the premise like that. Your argument either reduces to the circular "God is good because God is good" or the appeal to authority fallacy of "God is good because I say so." Neither is a valid objection to the dilemma.
The way this objection is most commonly phrased is "God is good because it is God's nature to be good." Again, this is either circular logic or an appeal to authority fallacy but phrased this way, you can see that it is an attempt to play a shell game with what is being claimed as the authority from which morality is derived. All you've done by raising this objection is to shift the claimed authority from God to God's nature, but now you have the exact same dilemma with God's nature that you once had with God. How do you know God's nature is good unless you develop a definition of good that is independent of God's nature? Same dilemma, different claimed authority.
The false dichotomy objection does not actually resolve the dilemma.
Obey!
Plato (or Socrates if you believe Plato was merely transcribing what Socrates said) proved thousands of years ago that no authority can possibly produce objective morality. So why then does every religion claim to provide what no religion can possibly provide?
This is pure speculation on my part because we can't go back in time and interview the people who invented religion, but I suspect it's just a cheap marketing gimmick. If you believe that your religion makes you a better person, then you will be less likely to question the man in the pulpit for fear that doing so will cause you to become a bad person. And if you are not part of a religion, well, you want to be a better person, don't you? Don't you want to become more moral? All you have to do is obey the man in our pulpit and you will be a better person!
In the Bible and the Quran, God asks Abraham to sacrifice his most beloved child. So he takes his son to the sacrificial altar (wait, why did they have a sacrificial altar in the first place?) and prepared to perform human sacrifice on his own son. At the last minute, God said "Haha! I was lying to test you! You don't really have to sacrifice your son! This proves that you are a good person!"
Do you see the sleight of hand here? The Bible is not teaching morality, but teaching Christians to confuse obedience with morality, exactly as the Euthyphro dilemma predicted.
In case you are confused, let me spell it out: human sacrifice is immoral. Sacrificing any human being to please a god is fantastically immoral. Killing your own son is immoral. Abraham was being asked to choose between morality and obedience, and the Bible/Quran tells us that obedience is the correct choice over moral good.
If you pay attention, there are other places in the Bible and the Quran in which characters are asked to choose between obedience and morality, and we are told that obedience is the correct choice, only it does not phrase it that way. The reader is invited to believe that obedience itself is moral. The reader is invited to confuse obedience for morality, exactly as the Euthyphro dilemma warns.
The Origin Problem
Morality cannot possibly come from the source theists claim. Authority simply cannot produce morality. If you know right from wrong, then that came from you. Not your religion, not your god, not your holy book, not from the man in the pulpit, but you.
Religion comes in with a sweet siren song and says "Just obey us and you will be good. Let us do the thinking for you and you will be good." While I'm sure this sounds appealing to some, there simply is no substitute for carefully considering the consequences of a decision before making it. There is no substitute for hard work and careful thought when it comes to morality.
Indeed, there are a variety of areas where theists are more likely to make immoral choices, from violent crime to keeping marriages intact to having unprotected sex. The good news is that theists are merely more likely to make bad choices in certain areas, which shows that many are, in fact, good people. This means that many Christians are not getting their morals from the man in the pulpit and are in fact doing the hard work of considering decisions before making them. I just wish more did so.