The argument from evil attempts to suggest the nonexistence of God on the basis of the existence of ostensibly gratuitous evil. If God existed, since he would be maximally good, he would not allow for the existence of gratuitous evils. Insofar as it seems there are many such evils, and we are justified in taking them to be gratuitous, we would have to conclude that God does not exist.
One particularly important line of response called "skeptical theism" attempts to undermine the claim to justification. The skeptical theist holds that we are not justified in judging any instance of evil gratuitous in such a way as would be incompatible with the existence of God, insofar as there is a huge intellectual chasm between ourselves and the divinity. Even if we can't see a reason for allowing the various evils that we see, it doesn't follow that God couldn't have a reason of which we would be ignorant. After all, he is omniscient and I'm not; he can see further and with greater detail than I or anyone else can. If anything, this would be expected.
The arguer from evil may rejoin that this kind of skepticism is difficult to contain; in fact it may lead to the undoing of morality altogether. For if God has a reason to allow some horrific evil to occur, then it may suggest to you that you are not under any moral obligation to help or assist or prevent evil. If God has reason to permit it to occur, wouldn't that justify you in allowing to it occur without dirtying your hands? Or worse -- by working to prevent evil, you may be working against God's will. In this way the suggestion is made, contrary to popular opinion, that it is actually theism and not atheism that is incompatible with morality.
This is nice and clever, but it doesn't ultimately prove too compelling. For I am not just a theist; I am a Christian. I believe that God has revealed himself and his will for humanity in the scriptures of my religious tradition, First and Second Testaments, as well as in the broader teaching of the tradition as it has been passed down through the ages. And God himself tells me to do good to all people (Gal 6.10), to be a neighbor in the manner of the good Samaritan (Luke 10.25-37), and so on. So my morality is not undone; God himself has told me what I have to do.
It is a misfortune and tragedy of so much contemporary analytic philosophy of religion that it treats theism neutrally, as if anybody were merely a theist and not also a Jew, Christian, Muslim, or whatever. If I were merely a theist maybe I would have a problem here. But I'm not, so I don't.
On the other hand, atheists help themselves to morality a bit unfairly. How in the world should an atheist know what is right and wrong? Her moral intuitions, the base of her moral judgments at the end of the day, are themselves largely the product of two thousand years of Christianization of the Occident. For the vast majority of human history, people didn't think minimizing harm, caring for all human beings irrespective of tribe or nation, etc., were important moral values. If there's been any progress it's been had because of the spread of Christianity, and atheist morality, if there is such a thing, is largely drawn from the Christian impact upon history.