The problem of evil is this:
1. God is all-good.
2. An all-good being would do everything in his/her power to eliminate evil.
3. God is all-powerful.
4. Therefore, evil should not exist.
5. We know that evil does exist.
6. Therefore, if God exists, he is either not all-good, or not all-powerful.
For the atheist, this is case closed: there is no God. Or if there is, he is either not all-good or not all-powerful, and what sort of God is that! Certainly not the God of Christianity, at any rate. On the other hand, for the theist, this is a challenge to come up with a theodicy - an explanation of why, if God were all-good and all-powerful, he would choose to let evil remain. In order to be an adequate response, this explanation must be consistent with itself and with the belief system of which it is a part, and also with the world as we know it.
The most obvious route for the theist is to qualify statement 2 above thus:
2. An all-good being would do everything in his/her power to eliminate any particular evil, unless doing so would result in a greater evil, or the loss of a good greater than the mere absence of the first evil.
This qualification would then therefore imply that the evil that we see in the world is in fact unavoidable, in order to achieve a greater good or to prevent a greater evil. The rejoinder to this is to ask: what is this greater good that evil must be tolerated in order to achieve; or what evil could be worse than that which we see around us every day?
One widely accepted argument in theistic circles is that there is a greater good: that of free will. In this view, God values one thing above all others, and that is man's ability to freely choose between good and evil. To put it starkly: a Hitler's ability to choose his own actions is of more value to God than the prevention of the suffering of concentration-camp-fuls of innocent people.
How the free-will argument explains the existence of evil is to say that for an agent to be morally free requires there to be the possibility of doing good, and the possibility of doing evil. We can put it this way: if an agent A is presented with a choice C between a good action a1 and an evil action a2, then if it is impossible for A to choose a1 or a2, then A is not morally free with regard to C. Thus for humans to be morally free agents, there must be the possibility of doing evil actions. Therefore there must be evil in the world.
The argument makes perfect sense right until that last sentence. For humans to be morally free requires the possiblity of committing evil; agreed. But that doesn't mean that the possibility need ever be realised. If the morally good choice is always the best choice, for the agent or for the population, then surely a fully-informed, totally free, intelligent, conscious agent will always choose the moral good; to do otherwise would make no sense whatsoever.
The trouble is, that's not the human race that you and I know. So what went wrong?
"Aha," says the theodicist, warming to the argument, "this is where the Fall comes in. You see, God created Adam and Eve perfect and morally free. They had a moral choice between a evil action (eating the fruit) and a good action (not eating it), but they chose the evil. So that's when sin entered the world. And now we're all sinners because we have a bias towards choosing evil."
Ah, it all makes sense ... hang on, I thought they were perfect and morally free. And I just argued that for someone in that position to do anything other than the moral good would be nonsense. So why did they choose the evil?
Answer #1: They were tempted
So? Before the Fall, humans had no bias towards evil. So there would be no irresistable urge to choose the wrong action. So temptation would have no power, because however attractive an option might seem in the short term, the perfect, morally free human would always choose the good, no matter what the temptation, simply because it's the better choice.
Answer #2: The Devil deceived them
Hmm. OK, we can't see any way perfect beings could choose evil, so let's postulate an essentially evil entity which temporarily assumes power over them and convinces them to do something which they would never choose to do otherwise... Leaving aside the question of where this guy came from, and how he became evil, where was God at this time? What was God doing letting this smooth talking, evil being wander about in his paradise? This being with the power to completely deceive innocent humans into doing something completely irrational and with such disastrous consequences? Because it stands to reason, given that perfect, morally free agents would always choose the best course of action, the Devil must have interfered with their reasoning processes. Which means that it was not their fault. Which means it was the Devil's. Which means, ultimately, it was God's, for letting him in in the first place.