Omnipotence
Omnipotence is unlimited power. It is a concept that we draw by comparison to limited powers.
A limited power is a collection of other powers which lack an unlimited power to remain held together. A human hand holds itself together in a good way unless some other power causes the hand to break internally or to become separated into a number of parts. That same hand, combined with the brain to which it is connected, has the power to make a hammer. But, since the hand already is not all-powerful, then the hammer, when applied to the hand, is more powerful in some ways than the hand. Only an unlimited power can hold itself together, or remain coherent, no matter what other powers are applied.
God is described as being omnipotent [1].
The controversies concerning how to understand or define omnipotence are resolvable in terms of the respective ontologies of power, all-ness, and logic.
Some atheists have argued that omnipotence is an inherent paradox, as it can be construed as self-contradictory. For example, suppose an omnipotent God were to attempt to create a rock so large that he could not lift it. This attempt must be either successful or unsuccessful. If it is unsuccessful, then this is a task which this God is not capable of, and thus the God cannot be truly omnipotent. If, however, the attempt is successful then God is now presented with a rock which he cannot lift. Again, this means that there is an action which the God is incapable of, and thus the God cannot be omnipotent.
The problem with the argument is that it poses a nonsense request (akin to asking if God can create a green smile or a fast tree), and then presents a false dichotomy (specifically, by way of positing an unexceedable power which is exceedable). There simply is no such thing, so the argument does not address what omnipotence is: a peerless agent. To ask if God can create a rock too heavy for Him to lift amounts to asking if He has the power to create something that's more powerful than Himself. It's an invalid, untenable question; like asking if infinity is finite, and then being unsatisfied when the answer is "no".
The only reason it is possible to come up with questions that seem to show that omnipotence is a paradoxical concept is by drawing from certain kinds of facts about limited powers. A human has limited powers, yet a human has the power to smash her own hand to bits with a hammer that she has the power to make. Clearly, this does not mean that her hand has the power to be smashed to bits; rather, it means that the things that make up her hand are limited in their power to hold themselves together. Only God is coherent no matter what other powers are applied to Him. And, only God is able to think straight no matter what facts anyone thinks can be applied to Him.
According to most atheists, omnipotence is nonsense. But, if omnipotence is nonsense, then there would be no way to construe it as a paradox. But, if, as even many atheists admit, omnipotence is a logically necessary (but, to them, incoherent) concept, then omnipotence cannot be an inherent paradox without making logic into a set of incoherent rules of reasoning [2][3].
God is also not capable of doing anything which is against His nature. For example, as God is by definition truth, then He cannot lie.
Skeptics further argue that it is impossible for God to be both wholly good (benevolent) and wholly powerful. The argument is that if God is both benevolent and capable of putting an end to suffering, then He would put an end to suffering. As suffering still exists, God must not be omnipotent or must not be benevolent.
However, this argument presumes that God would see the elimination of suffering as the benevolent thing to do. God has provided man with the gift of free will, but if God prevented all bad consequences resulting from our decisions, then He would be overriding our free will.
Further Reading
Essay: Omnipotence, and the logic of power
Essay: Why Is There Something Rather Than Nothing?
References
