The Euthyphro dilemma is a moral argument which presents Theists with two possibilities. Either God (or, more accurately, "the gods") is (are) capricious, or bound to a higher standard. Is a thing good because God says so, or does God say so because it is good?
In the former scenario, anything can be good. If God says murder is good, murder becomes good. If God says rape is good, rape becomes good. If God says theft is good, theft becomes good. Basically, if good is just whatever God says it is, God can justify anything, no matter how horrible.
On the other hand, if God only says a thing is good because it is good, there is something greater than even Him that defines morality. If God says "murder is evil", murder doesn't become evil, it's just that murder actually is evil, and so God is merely agreeing with a moral source that not even He can disagree with.
So, which is it? Is God capricious, or is there something with more authority than He has? The first issue with this so-called dilemma is that it doesn't really address the problem. Rather, it shifts it. If a thing was more authoritative than God, that thing would, itself, be God. Which, in a way, is true. Christianity holds to a philosophy called Trinitarian Monotheism. That is, we believe in one God that exists in three separate persons: The Father, the Son and the Holy Spirit. These three persons act in one accord, having the same will. In a sense, therefore, yes, God submits to the highest authority: God.
And that's actually the answer to this dilemma. Originally, the dilemma was posed to polytheists. Polytheistic gods are often divided, some even being capable of killing each other. They're born like men, they live like men, they fight like men, they die like men. The logical conclusion is that they have authority equal to men. However, even men are, in some sense, gods.
Let's use the stick man in the header image as an example. I drew the stick man. It's terrible, right? Except it's actually not. The stick man is perfect. Of course, you might disagree. You might point out that its head is wonky. You might point out that its arms are unequal. You might point out that one of its legs appears to be half erased. But the reason I can dismiss all of that and still say it's perfect is because the sole purpose of the stick man is to feature as an analogy for this article. Because I was able to use it as an analogy in this article, the stick man has fulfilled its purpose. The stick man is good because I say it is, and I can say it is good because it is good.
This is exactly how God, as Creator, looks at our world. God created our world for a purpose, God can tell when it fulfils that purpose, and when that purpose is not fulfilled, God can say that's bad.
This is not arbitrary. Theoretically, God could say murder or rape or theft etc. are good, if He designed the world to involve those things. However, because God designed life to be lived, sex to be enjoyed between a consenting married couple, and property to be a reward for diligent work, we can know that God won't just flip a switch and say "eh, enough with the thou shalt nots, go nuts". And this does not allow for a higher authority than God to exist, because God Himself set the standard. It is against God's nature to violate God's nature.
Therefore, the answer to the Euthyphro Dilemma is that good is good because it fulfils God's desires, and God calls it good because it succeeds in doing so. Just as it is possible for me to call my stick man good because it fulfils the purpose for which I designed it, God can call good and evil good or evil based on how well they conform to His will.
The question each of us needs to ask ourselves is how much we want to conform to His will. We've all ultimately cracked ourselves, so to speak. Every single one of us has done things God desires we avoid, or not done things God designed us to do. And what does man do when our creations fail us? Ultimately, we either burn them, or dig a giant hole and bury them. But are there not other options? Recycling? Upcycling? Repair? God prefers all of those things to just destroying us. And so God formulated a plan. Knowing ahead of time just how much we would fail, God put a rescue mission into operation. Rather than destroying us, as we justly deserve, God sent Jesus into the world to be born of a virgin, live a perfect life, and die a sinner's death. After that, our debt was paid, and all who believe Jesus rose from the grave will ultimately rise with Him. The choice is yours. Destruction, or resurrection.