The argument from evil (or problem of evil) is the argument that an all-powerful, all-knowing, and perfectly good God would not allow any--or certain kinds of--evil or suffering to occur. Unlike the logical argument from evil, which holds that the existence of God (so defined) is logically incompatible with some known fact about evil, the evidential (or probabilistic) argument from evil contends that some known fact about evil is evidence against the existence of God. For instance, one version of the argument contends that the biological role of pain and pleasure is much more likely on naturalism than theism (e.g., Paul Draper).
Other versions of the evidential argument concede that God could have a morally sufficient reason for allowing certain evils to occur--e.g., to ensure that some greater good is achieved as a consequence of an evil. However, proponents add, God would only allow as much evil or suffering as is absolutely necessary in order to achieve greater goods. But when we look at the world around us, we find prevalent instances of apparently gratuitous evil--pointless evils from which no greater good seems to result. According to proponents, the existence of apparently gratuitous evil provides strong evidence that God (as traditionally defined) does not exist (e.g., William Rowe).
For thousands of years theologians and philosophers have developed elaborate theodicies--responses to the argument from evil which retain belief in an all-powerful, all-knowing, and perfectly good God. According to the unknown purpose defense (UPD), God allows apparently pointless suffering for some reason that we can't comprehend. The free will defense (FWD) maintains that God has to allow the existence of some evil in order to preserve human free will (e.g., Alvin Plantinga, Robert Adams). Finally, the soul-making theodicy (SMT) contends that God allows some evil because it builds positive character in the victims or in others which outweighs the negative value of the evil itself (e.g., John Hick).
There are several problems with each of these theodicies, but I will only note the most serious ones. The UPD faces the obvious objection that if you have no idea what reason God has for allowing evil, then for all you know there is no justifiable reason at all for an all-good God to permit it. And even if the FWD and SMT were successful, they would still leave much apparently gratuitious evil unexplained. As William Rowe points out, when a fawn burns to death in a forest fire and no human being ever knows about it, this apparently unnecessary evil neither preserves human free will nor builds the character of human beings.