The so-called "problem" of evil is one of the oldest (and, by extension, weakest) arguments against the Christian faith. If God is all powerful, it is reasoned that He should be able to stop all evil in an instant, and prevent more from beginning. If He is all knowing, He should know exactly how to do this. If He is all loving, He should be more than willing to stop all evil. Thus, if evil exists, it is suggested that either God doesn't, or that He is lacking in power, knowledge, or love.
Of course, evil does exist, and so using this logic, atheists continue to assert that the God described in the Bible does not. This can be answered a number of ways, starting with the simple fact the Bible explains the origins of evil quite explicitly. Specifically, it points to us as the originators of evil. Yet, God loves us, and takes no pleasure in our deaths, preferring our repentance. It's also quite clear that God being "all powerful" does not mean beyond limits. The Bible ascribes a few limits to God, all stemming from His own nature. God "cannot deny Himself", meaning He is internally consistent. You may think of the law of non-contradiction as an attribute of God.
There are some very lengthy discussions we may have about free will, such as why did God give it to us, and will He remove it in Heaven? But for this article, let's simply point out that He did give it to us, and thus we are ultimately responsible for evil.
But atheists tend to have a different idea of evil than God, and even Christians. When they talk about evil, they really mean the things they don't like. Of course, sometimes that is aligned with God. God doesn't like murder, atheists don't like murder, so they ask why does God allow murder? But they're far less likely to ask why God doesn't prevent gluttony, or non-violent forms of sexual immorality, or sloth, or pride, or greed etc.
See, ultimately, we're all sinners, and so one source of evil is atheists. And they don't want to know why God doesn't stop their evil. They really want to know why, when they do evil things, He doesn't protect them from the consequences. Sin has both direct and indirect consequences. As they say, "play stupid games, win stupid prizes". You want to sleep around? Don't expect God to protect you from STDs.
Of course, not every bad thing is a direct result of sin. This is an error Jesus corrected when He and His disciples met a blind man in John 9:1-3: "Now as Jesus passed by, He saw a man who was blind from birth. And His disciples asked Him, saying, “Rabbi, who sinned, this man or his parents, that he was born blind?” Jesus answered, “Neither this man nor his parents sinned, but that the works of God should be revealed in him." So you can't just run around saying "evil exists because we're sinners", because although true, it's slightly more complicated than that. Yes, there is a correlation between sickness and sin, but it's not always direct.
Nevertheless, the correlation cannot be denied. When man sinned, God's "very good" world in which there was no death, or disease, or suffering, or evil, became marred and destroyed, as He withdrew His sustaining power, and cut us off from the tree of life.
But what is the solution to all of this? It seems self evident that atheism is not an answer. See, atheism has its own problem of evil: There is no source of morality in atheism. If there is no God, there is no objective morality, and so atheists are forced to come up with a new way to justify their own morals. There are many problems with this, ranging from the fact that atheists lack unity and foundation for it, to the simple fact that they are wrong.
See, ultimately, there is a God. Specifically, Yahweh, the Lord God of Israel. It is entirely possible, due to His grace, to live as if He doesn't exist, for His word says "But I say to you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who spitefully use you and persecute you, that you may be sons of your Father in heaven; for He makes His sun rise on the evil and on the good, and sends rain on the just and on the unjust." (Matthew 5:44-45). God loves atheists, and He has set the world up in such a way that it does not discriminate.
Nevertheless, He has also built the world on a foundation of wisdom. Thus, causes have effects. So, in this world, if you do good, you will likely reap certain rewards, and if you do bad, you will likely reap certain consequences. Ironically, however, evil tends to go through seasons of apparent prosperity. But it is temporary, for eventually, every man, woman, and child, will pass from this world and stand before God, who will judge us according to what we have done.
This can go one of two ways. For those who die without Jesus, it will result in eternal punishment. We have all sinned and fallen short of the glory of God (Romans 3:23), the punishment for which is death (Romans 6:23). Therefore, no atheist will enjoy meeting the God they have denied.
But God's solution to evil is far greater than taking full control of mankind, like robots, or even letting us run riot while erasing all consequences. Instead, what God does is allow us to have this life, in which we do sin, and do receive some degree of consequence. However, He also keeps His eye on us, protecting us from perfect destruction, and giving us His instructions as another guard.
But there's still the issue of justice to take care of. He can't just turn a blind eye indefinitely. What was wrong must be put right; every evil thought, word, and deed, must be accounted for. Now again, this may result in eternal punishment. That option is open to you. But we don't have to pay the piper, so to speak, because God has a better idea. He sent His Son to live a perfect life as a man, yet He died as a sinner should. By bearing our punishment, He leaves His reward free for us to claim. The Bible says "For He made Him who knew no sin to be sin for us, that we might become the righteousness of God in Him." (2 Corinthians 5:21). In other words, we effectively "swap verdicts". We, being sinners, can be treated as if we were righteous because He, being righteous, was punished for all our sins.
Thus, the solution to evil is far more grand than any atheist could imagine. It's not to kick down all our guardrails and blame God whenever that inevitably leads to disaster. It's to take refuge in our Creator, who forgives even the worst of our sins. Faith or failure. The choice is yours.