One of the most popular and perhaps most effective arguments against both the existence of gods and believing in gods is known as the "Argument from Evil." It's a popular argument because it's not one which requires a great deal of sophistication or philosophical education to understand. It's effective because even the weakest forms of the argument make a strong case that gods, or at least any beings that look very much like the gods people tend to believe in, probably don't exist.
Many people who end up as atheists are inspired to take a much harder look at their religion and their theism after being forced to face the problem of evil and suffering in the world. Thus, even if the argument doesn't disprove gods, it starts people down the road of questioning and skepticism.
Logical and Deductive Arguments from Evil
The earliest formulation of the Argument from Evil comes from the Greek philosopher Epicurus, writing in the early 3rd century BCE:
Either God wants to abolish evil and cannot,
or he can but does not want to,
or he cannot and does not want to,
or lastly he can and wants to.
If he wants to remove evil, and cannot,
he is not omnipotent;
If he can, but does not want to,
he is not benevolent;
If he neither can nor wants to,
he is neither omnipotent nor benevolent;
But if God can abolish evil and wants to,
how does evil exist?
This is a logical or deductive Argument from Evil because it attempts to show that the existence of God is logically incompatible with the existence of evil. There are many variations on this but there is also another category called the evidential or inductive Argument from Evil. Arguments of this type merely try to demonstrate that the existence of God is highly unlikely given the existence of amount of evil in the world.
Logical or deductive forms of the Argument from Evil are both the strongest and the weakest, depending on how you look at them. They are the strongest because they can decisively disprove the existence of God; they are the weakest because God has to be defined very narrowly and specifically in order for the argument to get anywhere. These forms of the argument disprove a particular sort of god, but not many others. Obviously the existence of a malevolent or weak god wouldn't be disproved here. The existence of the ancient Norse or Egyptian gods also wouldn't be disproved by this argument.
Evidential and Inductive Arguments from Evil
Evidential or inductive forms of the Argument from Evil don't try to show that the existence of gods is impossible, just improbable. This means that even if you accept the argument, you aren't forced to reject the existence of any gods; you are, however, forced to regard the existence of gods as highly unlikely, and therefore probably not worth believing in.
Such an argument might, for example, argue that a sufficiently benevolent and powerful being that warrants the label "god" would be able to at least reduce the amount of suffering in world — not eliminate it entirely, just reduce it. Therefore, the existence of any unjustified and unnecessary suffering indicates that such a being probably doesn't exist. Such forms of the Argument from Evil don't generally justify denying the existence of gods, but it does justify rejecting belief in the existence of gods and being an atheist.
Theodicies: Responding to the Argument from Evil
Theists recognize that arguments from Evil are among the most powerful atheological arguments that can be offered, both from a logical and an intuitive perspective, so significant effort is invested in countering them. Responses to such arguments are called "theodicies," or a vindication of the justness and holiness of God in the face of evil.
Theodicies in response to deductive forms of the Argument from Evil try to show that the existence of evil in the world is not logically incompatible with God's love and power. Common means for doing this are to argue that evil is necessary for the existence of free will, virtues, and other qualities we humans need. Theodicies in response to inductive forms of the Argument from Evil may have to argue that each instance of suffering is indeed justified and necessary — not an easy task because even a single unjustified instance of suffering is potentially enough to render the existence of a god too unlikely to bother with.