I have received several complaints that my last treatment of the Problem of Evil was too complicated, so I decided to simplify my arguments:
The problem evil arises when one affirms that (1) God knows all true facts, (2) he can bring about any thing good (that does not involve a contradiction), (3) that God is morally perfect, (4) and such a moral perfection means that God should eliminate evil as much as he can. If one affirms all of these then there is a contradiction because there should be no evil, because a good and powerful God would eliminate evil as much as he possibly can.
Not a Contradiction for Christians:
However, for most Christians this is not a problem because many Christians would reject that moral perfection means one should eliminate evil as much as possible. The reason why Christians reject this is because they think that God has morally justifying reasons for bringing about or permitting evil in the world. In other words, the sort of moral perfection that God has is compatible with evil because he has morally sufficient or justifying reasons for permitting evil to occur.
A Possible Atheist Response?
One atheist response might be to say that God could not or would not have a morally sufficient reason for allowing evil. But how do you know that it is impossible that if God existed he would have no morally justifying reason for the evil he has permitted? The atheist can say that they cannot think of any good reason for permitting evil in the world. But just because you do not know of any good reason does not mean that there is no good reason. I might not know an answer to a math problem, but that is different than saying that there is no answer to the math problem. So just because we do not know God's reason does not mean that he does not have one. In fact, it is unclear if he does or does not just by reason alone, so an unclear argument cannot be used to show that a position is unreasonable (especially since we have over 10 good arguments in favor of God's existence). On atheism, all we have is one argument that is unclear because either way we do not know whether or not God has a morally sufficient reason for the evil he permits. One unclear argument cannot outweigh 10 clearer arguments.
The Probablistic Problem of Evil
A weaker claim would be to say that evil makes God improbable because it is improbable that God would have reasons for all the intense evil that exists. But the problem with this is that God's thoughts are beyond ours, much like our thoughts are above and beyond those of an insect. So we cannot say that it is improbable because for all we know it could be that the morally sufficient reason God has is beyond our understanding and comprehension, much like astrophysics is above an insect's comprehension.
Two Possible Answers
Unfortunately for the atheist, it only gets harder from here, because theists have developed two good possible reasons for why evil might exist.
The Non-Reformed Response
The first one is the most common and it is called a free will defense (being Confessionally and Philosophical Reformed I reject this one). God cannot do anything that is logically impossible but he can only do what is possible. What is impossible is that God cause or make free creatures to do whatever he wants. So the only way to have free creatures is to not control and cause them. What God wants on this view is to bring as many people into a saving relationship with him by their free will and since free will is involved human beings can freely choose to hate God and do evil. What explains evil things that occur is the free will of demons and humans which cause evil, but God does not cause evil. God lets them choose freely so as to bring about as many people to a free loving relationship with him as possible so that he can give them eternal life (there is no world God can create where people only do good on this response). This defense was developed by Alvin Plantinga and the consensus in philosophy is that this argument shows that logical problem of evil is unsuccessful.
My Reformed Response
The next possible defense was developed by Calvinists thinkers like myself. Calvinists believe that God is in control and ordains everything so there can not be free will as traditional understood by many. In this view God ordains humans to choose to sin so that he can show his love, grace, and mercy through Jesus Christ. Grace and mercy can only exist with sin because if something is sinful there is no reason to merciful and gracious. When evil is in the picture God can show his love even more to human being by loving them through Christ Jesus even though they are sinful. I show my love for my dog charlie even more so when he does bad things, if he does good then I simply am showing my love to him and not showing my unconditional love for him. Furthermore, with evil in the picture God gets to show his hatred and punishment towards sin. God shows his glory also by finally punishing and eliminating all evil in the end. Evil also allows God to redeem us by the most perfect picture of love, mercy, and justice which is the Crucifixion of the Lord Jesus Christ. All of these great things about God get maximized when evil is in the picture and so it brings God's Glory and hence this could be another morally sufficient reason for why evil exists.
(This was meant to be a simple, concise overview of the Problem of evil and not a complete picture of my own view).