For argument's sake, assume that there is a God. Then establish what we all mean by evil. Evil is anything negative that occurs principally, though not exclusively to human beings. There are different type of evil such as natural evil (such as volcanoes and floods) or moral evil (murder or theft, for example).
The problem of evil is a problem because of the following:
God is all loving.
God is omnipotent.
Yet evil still exists.
In theory, only two of the statements can be correct. If God was all loving and omnipotent then evil would not exist, yet it does. God might not be all loving and so let evil exist, or he might not be omnipotent and so can't stop most of the evil that exists.
This, however, is not a satisfactory answer for most people who want to accept their particular God. They want an explanation that removes all blame from God. These arguments are called theodicies, and one that established the Christian tradition for centuries runs something along the lines of: we (the human race) were created in a perfect world, but we chose to turn away from God and so evil is good things gone wrong.
There is one fault with this and it is that it relies on the idea of people being continuous beings before evil things happened, which science and the evolution theory disprove.
If God exists, Why Do We Have Evil?
Evil may exist to give us free will, for example if there was no evil there would be no free will because there would be no consequences of our actions.
Evil may be a throw back to when people killed each other as a form of natural selection. In today's society this is not acceptable and so moral evil could be stopped by educating people.
If evil is a way of giving us free will, then it is placing a limit on God's power, thus making him something other than omnipotent.