Omnipotence refers to an agent which is axiomatically over all other agents. Abstractly, such an agent is said to have unlimited power, or, omnipotence. This abstraction, called the concept of omnipotence, is a concept that we draw by contrast to limited powers (limited agents), and by the possession, or instantiation, of every kind of power 
Coherence vs. incoherence of omnipotence
Prior to any act, divine omnipotence is perceived as coherent with itself. In other words, omnipotence, as a concept, is immediately coherent.
The immediate coherence of the concept of omnipotence is how we even can get the sense that omnipotence is paradoxical: it's immediate coherence is (mis)interpreted as a limitation upon it. Only by allowing that it may apply its own agency upon itself, and, thus, upon the fact that things can be identified, is this supposed limitation 'remedied'. Know-ability is a function of the fact that there is a connection, or correspondence, between the knower and the knowable. But, the position that the immediate coherence of omnipotence is a limitation upon omnipotence is tantamount to the position that the possibility of knowledge constitutes an agent which is distinct from, and superior to, an omnipotent agent. In other words, this position is equivalent to the statement: "rationality is not co-extensive with omnipotence, despite that omnipotence is known immediately to be coherent".
The Paradox of the Stone
The paradox of the stone is the common argument given by atheists (and some theists, such as Peter Geach) to attempt to show that the concept of Divine omnipotence is incoherent and thus non-existent. If God can create a stone too heavy for him to lift, goes the argument, then God cannot do something, namely, lift the object; if God cannot make such a stone, then there is still something He cannot perform and thus is not omnipotent.
There are several ways to answer this paradox, but the most common is to say that omnipotence does not entail the ability to actualize logically incoherent propositions such as 1+1=3. This is not a limitation or incoherene in God's omnipotence, because 1+1=3 and a rock too heavy for an omnipotent being to lift are not things. In other words, it is the propositions themselves that are incoherent, not God's omnipotence. Since these propositions don't actually describe anything (because their descriptions involve logical contradictions), there is nothing for God to actualize.
Another less conventional way to answer the paradox is by saying that if God wanted to, He could take on human form and zap a rock into existence above His body, thus being too heavy for Him to lift. But the problem is, of course, that the ability to make a rock too heavy to lift is not an exercise of power. It comes from a lack of power which God does not have, just as sin is an example of a lack of goodness in humans which God does not have.
And thus the paradox of the stone fails.
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.
The central components of the debate
The controversies concerning how to understand or define omnipotence in regard to the nature of its relations to other things are all resolvable in terms of the respective ontologies of power, all-ness, and logic. Since the concept of omnipotence is coherent prior to any task, any task which does not abide the concept is precluded by the concept. Of course, some tasks, as such, are difficult to understand, while others are difficult to rightly understand in the first place.
Details of the debate
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. But, some people misunderstand their own thinking in regard to the concept when the question is answered with "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 were pure nonsense, then there would be no way to construe it as a paradox: Pure nonsense has no meaning whatever, such as any nonsense word that does not include sensible words (sgfdpfkp). So, there must be some sense to omnipotence to begin with. Furthermore, 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. Some atheists admit that omnipotence is coherent, though they reject the idea that it can apply to a sentient agent.
The omnipotence of God
God is not capable of doing anything which is against His nature. For example, as God is by definition truth, then He cannot lie.
Skeptics argue that it is impossible for God to be both wholly good (benevolent) and wholly powerful. This argument states 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.
- The Paradox of the Stone; C. Wade Savage; The Philosophical Review, Vol. 76, No. 1 (Jan., 1967), pp. 74-79
- Omnipotence and Quantification; Tersur Aben
- Revelation 19:6
- Some Puzzles Concerning Omnipotence; George I. Mavrodes
-  Reconsidering Absolute Omnipotence, by Louis Groarke
- Omnipotence Paradox: Overview; Wikipedia article