I will speak to the origin of evil below. For now, I am going to address the question itself, and why it actually demonstrates the illogic of atheism, not Christianity.
Mr. Atheist, what is evil? You probably answer something to the effect of "whatever hurts people." And I would certainly agree that hurting innocent people is evil. But then, on what basis do you decide that hurting people is evil? A cannibal would be fine with hurting people, because doing so feeds his family. On what basis would you say that he is wrong? Both Stalin and Hitler believed that shedding rivers of blood was good, because (in their minds) doing so benefited a greater number of others. Were they wrong? On what basis do you say so?
Those questions are my rhetorical means of making this point: a materialist worldview has no basis for deciding right and wrong, good and evil. We are, in the atheist universe, just DNA seeking to sustain its existence. What an atheist views as evil derives from his cultural heritage of biblical theism, i. e., Christianity. This is called "precept stealing." Therefore, for the atheist to ask the question with which I started this post is for him to admit - unconsciously, I admit - that Christianity is true. That is why I
To the Christian who has been befuddled by the question, I will now turn to answering it. The problem is the assumptions that undergird it. It assumes that I, or you, or whoever - i. e., a finite human being - is able to judge what is good, above the ability of the omnipotent, omniscient, and omnipresent God. It assumes that I can know every possible way in which an act can be good, but He cannot. Or, put another way, it assumes that He cannot have a perspective that shows the goodness of His acts in a way that I cannot perceive.
Let me give a hypothetical illustration: imagine a wreck, in which a drunk driver kills a mother and her two children in another car. I am sure that we agree that this would be an awful thing, so how could that be good? Well, what we might not be aware of us is that there was a school bus full of kids the next block away, which would have been hit by the drunk driver, if he hadn't been stopped. When we accuse God of allowing evils in our lives, those are the things we cannot know. What would have happened otherwise, that I cannot foresee? To ask why an all-good God makes the choices He does would require us to have the same omniscience that He has. Why does an all-good and all-powerful God allow evil? To prevent greater evil. It is impossible for us to know any more than that.
The source of evil can be found only in the Bible. Atheists might ask the question, but they cannot answer it. The Bible, however, reveals that God created a world without evil. Yet He also placed a choice before Adam and Eve in that world. They could have eternal life without evil - the condition in which they were created - or they could choose to eat of the Tree of the Knowledge of Good and Evil and die. We all know the choice that they made: they ate of the fruit and brought disease, futility, hardship, and death, upon themselves and upon their posterity. Mankind chose the kind of world for which the atheist seeks to blame God. That is why there is evil in the world. You can read this in the Bible, in the book of Genesis, the first three chapters.
The wonderful thing is that the story doesn't end there, because Christianity doesn't answer just the question of the origin of evil, but also how it will end. That is another answer that cannot be found in atheism. Through the Gospel, as sinners are converted, change our lives, and then change our world, God is creating a New Heaven and New Earth, where there will be no more evil. This is described in a number of places, but start with Isaiah 65:17-25 and Revelation 21:1-4.
You can find other thoughts I have had on this question here and here.