The omnipotence paradox is generally summed up by the phrase or some form of the phrase "Can God create a rock so heavy that he can't lift it?". Either God can or can't create the rock. If he can create a rock so big he can't lift it then he's not omnipotent because he can't lift it. If he can't create a rock that big in the first place then he's still not omnipotent.
The omnipotence paradox is related to the irresistible force paradox which asks the question "What happens when an irresistible force meets an immovable object?" The latter is self-contradictory since an irresistible force would deny the existance of an immovable object and vice versa. Thus, to say that an irresistible force could ever meet an immovable object is absurd. The same logic applies to the omnipotence paradox. If God is omnipotent then it's not possible for a rock to be too heavy for him to lift. If a rock is too heavy for God to lift then he's not omnipotent. Thus, the omnipotence paradox is absurd and, therefore, it doesn't make much sense to use it in an argument.
Accidental vs. Essential Omnipotence
The omnipotence paradox may be resolved by stating that God is either accidentally or essentially omnipotent.
An accidentally omnipotent God could resolve the paradox by creating a rock so large that he can't lift it and thus, ceases to be omnipotent. However, one could argue that it's hard to tell if a god of this sort was ever truly omnipotent or just in possession of great power. Also, not many Christians would want to use this resolution since it implies that God is only omnipotent some of the time.
Essential omnipotence states that it is impossible for the god to be non-omnipotent and that the god can only do that which is logically possible. With this resolution the god is intrinsically unable to create a rock so large that it can't be lifted yet it retains its omnipotence the whole time. One could say that god is unable to do the logically impossible and is, therefore, not omnipotent but saying that someone could do the logically impossible is inherently absurd and shouldn't be used in an argument.
Even if we grant that God can do all things except those which are "logically impossible", there still seem to be limits on God's abilities in practice, though not prohibited by logic. Can God act in a way that is evil? By definition, God is supposed to be omnibenevolent, so the answer should be no. Yet people can do many things that are evil, so clearly being evil is logically possible, but God supposedly cannot do it.
God's Free Will
Apologists may respond to the above argument by stating that it isn't that God can't do evil, it's just that he always uses his perfect free-will to chose not to. This, of course, raises the question of why God didn't create humans the same way. It would be a good solution to the problem of evil.
Another answer to the above argument is the assertion that being good is "God's nature," and He can only do things that aren't against His own nature. At this point, the concept of omnipotence starts to get completely fuzzy. "God's nature" appears to be defined as "the things that God is capable of doing." In this case, the claim of omnipotence is nothing more than the tautological statement, "God can do all things that God can do." Under this usage, people are also "omniscient."
The fundamental problem may be that the concept of being all-powerful is ill-defined and lends itself to equivocation.