What is a Christian explanation on why evil and suffering exists?

Because of the original sin of disobeying God and the subsequent fall. The world and everything in it is not as it should be.

From the book of Romans:

For I consider that our present sufferings cannot even be compared to the glory that will be revealed to us. For the creation eagerly waits for the revelation of the sons of God. For the creation was subjected to futility - not willingly but because of God who subjected it - in hope that the creation itself will also be set free from the bondage of decay into the glorious freedom of God's children. For we know that the whole creation groans and suffers together until now. Not only this, but we ourselves also, who have the firstfruits of the Spirit, groan inwardly as we eagerly await our adoption, the redemption of our bodies. For in hope we were saved. Now hope that is seen is not hope, because who hopes for what he sees? But if we hope for what we do not see, we eagerly wait for it with endurance.
 

Roman Catholic Answer

The definition of evil is a lack of God. God created everything in the world, and called it good. Anything that has being or existence is good. God created angels and men with free will so that they could choose. They do not always choose God, when they choose something other then God, they are committing sin and evil enters in. Think of it like this: if God was the sun shining in the sky, now, turn around, away from the sun. What do you see? A shadow. The shadow is NOTHING, it is an absence of light. That is what evil is. The devil, himself, was created good, it was only when he turned away from God and in on himself that he became evil.


The only one who has existed forever is God; he is the only beginning of everything, and he created only what is good. The Bible reveals to us a God who is pleased because he has created everything beautiful and good, with the goal of life and happiness.

Evil entered the world when God willed to create beings of extraordinary greatness because he made them intelligent and free. Freedom, both for angels and for men, is an irreplaceable greatness. Evil came into being because first an angel and then man abused this gift of God. Therefore, evil did not always exist. It originated when some of the angels rebelled against their Creator. Even then, God manifested His divine mercy and wisdom, because, even though he tolerates evil--he never rejects his creatures--nor does he take away their power to harm others--he is able to bring about good from evil. Therefore, even though diseases, suffering, persecutions, perfidy, and all the evil in the world did not originate with God, he can use them for our sanctification and, therefore, for our own good.