So much depends on your preconceived principles of faith. I couldn't give the same answer to someone who doesn't believe in God as I could to someone who does. Furthermore, what is your conception of God? What are God's defining characteristics?
Atheists have no problem with this question, of course. Some smart-aleck once said that atheists have no problem with the existence of evil, only with the existence of everything else.
I'll let that go, for now.
Assuming that you believe in something God-like, then you still don't necessarily have a problem with the existence of evil if you believe in one of the following things:
1. God is not omnipotent.
Even if you believe in a single deity, you can believe that God is not omnipotent. In which case, you can assume that evil things can happen beyond God's control.
By the way, this is the answer given in the book "When Bad Things Happen to Good People", by Rabbi Harold Kushner.
2. God is not omniscient.
If you believe that God is omnipotent, you still may not believe that He is all-knowing. Perhaps bad things happen and the most He can do is react to them. Alternately, you may believe that God "wound the clock", so to speak, and then left, so that He doesn't have direct involvement in our worldly affairs.
3. God is not omni-benevolent.
Even if you believe the first two, evil is not a problem for those who do not believe in God's benevolence. I believe that that is the case for many people who suffer, or who view the terrible things that happen around the world.
If you do believe all these things, then you have to answer how an all-powerful, all-knowing, and good Being can let terrible suffering occur.
Explanations I Can't Accept
One explanation that I often hear is "because you've sinned". Rabbis and Preachers love that one. After all, it is hard to prove otherwise. No one can say that they're perfect.
I find this explanation to be pretty pathetic. First of all, you can simply look at thousands of people whose lives are roughly equal in virtue and see that the pain and pleasure allotted to each is wildly disproportionate. They can say that the sufferers must have sinned "in their heart", which is not only unprovable, but highly improbable.
In any case, if you're Christian or Jewish and believe in the Old Testament, you're completely contradicting the book of Job, whose entire premise is that bad things happen to good people, even when they haven't sinned. Job's three friends believe that he must have sinned, and both Job and God reject this.
Furthermore, a tremendous amount of suffering happens to babies and children, who cannot possibly have sinned before they were born.
The explanation given in Job is that God's ways are mysterious. He made the Earth and Heavens, and so it is not possible that you will ever really know the answer. While possibly true, to an extent, this explanation is not entirely satisfying.
Another explanation is that a person may have sinned in a past life. For this to make sense, you have to assume that suffering is not "correction" - i.e. used for instruction - but "absolution", in that it somehow cleanses the soul.
I find this a rather inefficient system. I can't see the point of having another life just to suffer for a previous one. Furthermore, the non-obviousness of this system is a stumbling block for human understanding. We have to make sense of it retroactively; otherwise, we couldn't hold our faith. But we only hold our faith because of this system. That's rather circular reasoning to me.
A similarly unsatisfying explanation is that the more suffering you undergo in this life, the less you will receive in "the world to come". This is another example of reasoning invented to explain what we cannot otherwise cope with.
Another explanation is that we suffer so that we are inspired to choose God as our savior. God is showing us what free will allows us, and why it's bad. Again, the number of babies and children who suffer without ever having the opportunity to do this, not to mention the suffering that occurs after choosing God, seems to preclude this as a possibility.
Here are my thoughts.
In one of my wife's papers, she wrote about Emmanuel Levinas:
According to Levinas, it is "disappointment", the absence of God, which forces the individual to created a space within for God as Other. A separation, a distance is necessary. The individual is thus not enclosed in a totality which dissolves the self, but allows him to give himself over, turn towards the other, his fellow man. In other words, "ontological absence [of God, the Other] means ethical presence [for the other]".Even better, Rachel once phrased this as: "ontological absence necessitates ethical presence".
Which means that God perforce gives humans something to do by absenting Himself, just a bit. Without evil to overcome, we are nothing but empty vessels.
Can an omni-benevolent God create evil and suffering? Yes, if by doing so even more good comes as a result: human compassion, human action, human striving.
Can this really explain a child born to twisted agony, destined for a short life of starvation in Africa? Or the Holocaust? By virtue of God's absence, we must act together to stop these things from happening. We have to invent the science to help the unborn, the economics to share the food, and the politics and militaries to stop the tragedies. We cannot rely on God to do this for us.
But an explanation is not an excuse. That's one of the points, I think. God's examples of prophets are Abraham, Job, Moses. These guys railed against, argued with, and condemned God. They wouldn't let God off the hook. And they took action.
If you don't like evil and suffering in the world, fight against it and, while doing so, condemn God for it. Bring God to task.
More on the subject of theodicy. And here's an entire blog devoted to the subject.
Update: Over shabbat I thought of one more important point to make on this issue.
I didn't really answer how my interpretation of why evil/suffering exists can explain a small child born into pain and suffering who leaves the world before cognition, something which the other explanations fail to address satisfactorily.
I don't have an answer for the pre-scientific world, but for today's world the answer is that we have failed the child, at least partially.
Where is the medicine and nutrition that didn't make it to the parents before the child was conceived? The screening before it grew in the womb? The surgeries and gene-splicing that could have cured the infant before it was born?
Our science and our social programs have a long way to go, in part because so much of our world's attention and resources go to other issues, such as war and other such nonsense. In a sense, war's cost can also be measured indirectly, by our inability to give our fellow humans the attention and solutions that they need.
I don't know if, suddenly, the world were to act in complete concert that we could solve all human suffering, but I bet we could make a big dent in it.