The human experience of evil is a major problem for polytheists, dualists, and atheists. It's also a problem for monotheists, but one I think they can address best.
Polytheists do not face the same problem (since they can believe in conflicts between gods), but as far as I am concerned polytheism is a trivial answer to the problem of evil.
What clearly does work as an answer is radical dualism, i.e. the view that there are two opposing first principles. Variations on this theme which ascribe evil to a created devil figure do not really provide any answers. Note that while some historical dualisms have posited evil as a conflict between spirit and matter are not the only possible versions. In some forms of Zoroastrianism the Good Principle creates the physical world as an arena for the battle between good and evil. The problem with dualism is that while it does explain a conflict, it does not explain why one principle is good and the other evil. To provide such tags requires a level of meaning more fundamental than the two first principles, which are thus not first principles.
Atheists have perhaps a greater problem with evil. Let me be clear, many, perhaps most, theoretical atheists are highly moral people. Indeed, I think the whole argument that one needs to believe in God to be a good person is off kilter. In practice I suspect habit is the major guide in life to being a good person, and a habit of virtue is the way to be a good human being. (Such a position is entirely consistent with theories of behaviour in many varieties of Catholicism and Judaism). But faced with raw evil, atheism has nothing to say. For atheism, theoretical ethics must be nothing more than a combination of innate instincts and contingent social constructions. It is quite possible from such a point of view to condemn certain acts as destructive of social goods, or to come up with a theory of law. It is much more difficult to say certain acts are absolutely wrong. And it is impossible to see as evil the impact of natural forces. Atheists simply cannot coherently say that pain and suffering are evil.
The problem of evil for monotheists is this:- If God is an all powerful then God could create any possible world. There is a possible world without pain and suffering (and even if there is no possible world consistent with free will that has no pain and suffering, there is surely a possible world with much less pain and suffering than this one.) If God could have created a better world than this one, but did not, then God made evil. If God was not able to create a better world, then God is not omnipotent (and thus not, in an important sense, God at all).
There is no easy answer to this, but I think monotheism does provide better answers than the alternatives presented.
First, with God as the unique ground of being, it is possible to identify goodness as identical with Godness. Goodness is simply that which is accord with God. Evil is what is opposed to God.
Second, there is at least one theoretical explanation for evil possible with theism which allows both that human experience of evil is real and yet consistent with God as the ground of being. This is Augustine's notion that evil represents lack of actualisation of God's will. I don't find it entirely convincing as an argument, but it is better, I think, than any other.
Third, actually existing religions try to grapple with evil in an honest way. The Book of Job in the Jewish Bible (i.e. the Christian Old Testament) in a novelistic form addresses the reality of evil as it occurs in a human life. Job presents evil as real, random (i.e. not a proportionate response to some human sin), and ultimately inexplicable. It's only real statement is that the will of God is beyond human comprehension. This is scant comfort, but rather more comforting in fact than an easy answer. Christianity could be said to provide an explanation for evil with it's idea that evil is the result of sin. Now, I do think that the current band of published atheists such as Dawkins avoid the quite real issue of human sin, but I don't think myths of Satan tempting Eve provide much of an answer either. What is striking in Christianity is that it's central story - God incarnate as Jesus of Nazareth - does at least involve God in all the muck and mess of evil in the world.
The Christian version of monotheism does not provide an answer to the problem of evil, any more than does the Book of Job, but it does address the reality of evil (and does not try to cast it as an illusion like eastern religions), and states that God is involved in the mess. That, to me at least, provides some comfort.