The so-called "Problem of evil" is a strange argument, because it can effectively be stated as "the world we live in is exactly like the Bible says it is, therefore the God described in the Bible does not exist".
This strange reasoning comes from a misunderstanding of the term "omnipotence". Atheists create a straw man version of God, insisting that He should be able to do all things, including the incoherent. A round triangle is illogical, yet should not be outside of God's abilities according to these atheists.
Of course, the obvious flaw in this version of the argument is that you cannot argue against that which transcends logic. If God can be so incoherent as to create a round triangle, He can be so incoherent as to create evil and make you 100% responsible for it. Thus, the straw man is actually "stronger", at least in that sense, than the real God. As it happens, however, the Bible does not present an incoherent God, but one who operates in a specific way. It explicitly tells us that God can neither lie, nor deny Himself (Titus 1:2; 2 Timothy 2:13).
Because God is a coherent God, it is entirely possible for Him to co-exist with evil, simply because He has created a thing called "free will". Free will is what separates us from our machines. Can a computer love you? Can a car show you gratitude? Can you reconcile with a rebellious iphone? Of course not. Relationships with machines just aren't special, because they're not real. It's one step away from making out with your own reflection.
Free will makes real relationships possible. But it also comes with a setback: It also makes negative relationships possible. The ability to choose good allows us to choose evil. And that is exactly what we did.
When God created the heavens and the earth, they were very good. The world we see today is a mere shadow of it. Today, we have sickness, but there was no sickness then. Today, we have heavy work, but work was not so burdensome then. Today, we have sorrow, but there was no sorrow then. Carnivory, parasites, the pain of childbirth, genetic mutations, thorns and thistles, these things were all completely absent in the original creation as it existed at the end of the 6th day. Adam even walked with God in perfect fellowship.
But God gave Adam a very simple command: Do not eat from the tree of knowledge of good and evil, lest you die. And of course, Adam ate. Thus, God fulfilled His promise. The very ground was cursed. In fact, scripture tells us that the entire creation is cursed because of sin. Even out in space, where there is no life, there is the curse of sin. We expect to see evil, not because God doesn't exist, but because we exist.
This even answers why God, who could easily destroy all evil in a heartbeat, does not do so. See, He hates evil, but loves us. There is a popular question we must all ask ourselves: If God destroys all evil at 12:00, where will you be at 12:01? For the atheist, the answer is Hell. An eternity of separation from God, facing the due penalty for all sin. But this isn't what God wants for us. Scripture tells us that the death of the wicked brings God no pleasure. Rather, He prefers the wicked to turn from their sin and live.
Thus, God did something far more amazing than just stopping evil. Long before the first couple sinned, God planned the Gospel. When the time was right, He sent Jesus to the Earth to live as a man, a sinless life, after which He died on the cross. There, He suffered the full penalty for sin, meaning everyone who confesses Him as Lord, and believes He rose from the dead, will be forgiven for their sin, and rewarded for His righteousness. The result: Eternal life in a Kingdom far greater than even Eden.