Also known as the Argument from Evil.
The Problem of Evil is an extremely well-known and constantly rehashed theological/philosophical argument. It is intended to prove that the idea of the Christian God contains a contradiction. It is often generalized to refer to the 'Judaic-Christian-Islamic God' (G-C-I God), as all three gods share the relevant characteristics.
It is generally assumed that the J-C-I God is both omnipotent and infinitely benevolent. If you do not hold these to be true, the argument will not hold.
(1) There is evil (pain, suffering, etc.) in the world.
(2) If God could avoid evil, he would.
Therefore:(3) It cannot be true both that God could avoid evil and that he wants to avoid evil.
Therefore:(4) Either God could not avoid evil or God did not want to avoid evil.
(5) If God could not avoid evil, then God is not all-powerful (omnipotent).
(6) If God did not want to avoid evil, then God is not all-good (benevolent).
Therefore:(7) Either God is not omnipotent, or God is not benevolent.
This is one of the strongest arguments against the J-C-I God as it is commonly conceived.
One of the strongest objections against this argument is that evil must exist to allow us to have free will, but this objection does have some flaws.
The objection generally runs that free will is good AND that if we could not do bad things we would not truly have free will. Therefor, in order to maximize good God must allow free will and allow us to do Bad Things.
My favorite answer to this objection is that there are many things that Humans cannot do, no matter how hard they will it. For example: If I decided that I wished to kill all Jews, I would be lucky to kill 10,000. Hitler did wish to kill Jews, and killed quite a few more than 10,000. He did not have more free will than I did. He was just 'lucky'. Therefor, some evil is not the result of free will, it is only the result of 'luck'. I argue that God did not have to allow 'luck' in order to maximize good.
God gives us free will, but he limits in many ways. I have some choices, but not others. I can kill someone, but I cannot fly; I cannot read minds; I cannot live for 500 years. God denied me all of this. Why couldn't He have also denied me the ability to kill? In what sense is this ability needed in order for me to have free will?
Does it seem that we have the best of all possible range of possible actions? The Christian God does have control over Everything. Why did He decide killing/torture/suffering was necessary for free will? It may be necessary for us to sin in order for us to gain any moral merit from our free will, but there are other, happier, ways to sin.
Perhaps an even stronger objection is that, in theory, God has free will, and yet He does no evil. It is debatable as to whether or not God could do anything evil. This suggests that evil and free will are not necessarily linked.
Another interesting objection to the Problem of Evil is that God is inscrutable, and trying to explain (or, horror!, criticize) his actions is a fool's game.
There is also a severe, although perhaps not fatal, problem with this objection. Most of us sincerely, absolutely, and without a moment's hesitation believe that killing, rape, torture, and all that stuff, are BAD. And furthermore, we believe that people who willingly allow this sort of Evil are also bad. If it is Good for God to allow this sort of thing, that indicates that we are wrong, and that Evil is actually Good (at least sometimes).
Trying to argue that the ultimate standard in Good is inscrutable and incomprehensible leads us to conclude that it is impossible to know how to be good. Fortunately, we are left with the commandments to Love Thy Neighbor and Do Unto Others As You Would Have Them Do Unto You, which tell us how to be good, even if they are inscrutable. This leaves us, of course, in the position of weighing a moron who double parks in front of you as being as much of a sinner as is a murderer; they are both breaking the Golden Rule. And if God (and good) truly is inscrutable, we cannot question this.
It also means saying that God is "infinitely benevolent" looses all of its meaning. If we cannot understand the way in which God is good, then we cannot understand the term 'benevolent' as it applies to God. And moreover, the phrase "God is good" is unintelligible -- the same as saying that "God is xckdusa". It surely means something important, but we can't understand what that might be.