The argument from incoherence
It is widely thought that it is impossible to disprove a universal negative. For example, you could never prove that unicorns do not exist. The only way to know for certainty that unicorns do not exists is to be practically all-knowing. After all, unicorns may exist on some remote planet in some distant galaxy that we will never be able to explore.
But it turns out that there are some universal negatives that can be disproved. For example, you can disprove the notion that married bachelors exist just by demonstrating that "married bachelor" is a contradiction in terms. One cannot be both married and a bachelor.
The argument against God from incoherence is an attempt to disprove the existence of the Christian God by demonstrating that the essential attributes of God are contradictory in some way. If the Christian God is necessarily all powerful, all knowing, and perfectly good, and if being all powerful somehow contradicts being all knowing, then it's impossible for the Christian God to exist.
I'm not going to go through all the various attempts to demonstrate an incoherence in the concept of the Christian God. I just want to talk about one way that I've seen because it comes up a lot. It's one of those street objections you hear.
Can God create a rock too heavy for him to lift?
That's usually the way it happens. But lemme unpack that a little. Remember, the Christian concept of God entails that God is all-powerful, which supposedly means he can do anything. Well, this question reveals an incoherence in the concept of being "all powerful." If God is able to create the rock, then he would not be able to lift it, which means he is not all powerful since there is something he can't do. But if God is not able to create the rock, then he is not all powerful because, again, there's something he can't do. So whether you answer "yes" or "no" to the question, you find out that there's something God can't do, which means God cannot be all powerful. "All powerful," is self-contradictory, and can't be instantiated in any possible world, which means it's impossible for the Christian God to exist.
Christians usually answer this objection by saying that being all powerful doesn't mean God can do anything whatsoever, no matter how incoherent or irrational. They just mean that God can do all things logically possible. It is not possible for God to create square circles, married bachelors, or to exist and not exist at the same time and in the same sense.
A similar point might be made about God's being all knowing, though I never heard it brought up that much. One might ask, "Can God know something he doesn't know?" Well, if he knows it, then he's not all knowing, because there's something he doesn't know. But the only way he could know it is if there actually was something he didn't know, which would entail that he is not all knowing. So either way, it's impossible for God or anybody to be all knowing.
But hopefully it's obvious that you can't know something that isn't true. Knowing something entails that it is true. God can't know that the earth is flat, for example, because the earth is not flat. So it is no strike against his omniscience if God happens to not know that the earth is flat.
In the same way, it is no strike against God's power if he is unable to perform an incoherent act, such as creating a rock too heavy for an all powerful God to lift.
But suppose you've run into an atheist who is a little more sophisticated than the atheist-on-the-street, and he insists that the problem isn't that the notion of God creating such a rock is incoherent, but that the notion of being all powerful is incoherent. He objects to the Christian response by insisting that "all powerful" must mean God can do all things whatsoever, and not simply all things logically possible. And since doing "all things whatsoever" is incoherent, the Christian God cannot exist in reality.
There are two ways to respond to that.
First, let's just assume, for the sake of argument, that being all powerful does mean that God can not only do the logically possible, but he can also do the logically impossible. God can do all things whatsoever. In that case, all objections to God go away. If it turns out that we find an incoherence in God, that does nothing to prove his non-existence since we've already stipulated that he can do incoherent things. If we say that, yes, God can create a rock too heavy for him to lift, we can then go on and say that God is all powerful anyway. Being all powerful allows God to engage in logical absurdities such as being all powerful even though there are things he can't do. God can do what he can't do, he can know what he can't know, and he can even be all good and all evil at the same time. If by being all powerful, Christians mean that God can do all things whatsoever, then it would be impossible to ever disprove the Christian God.
Second, what should not escape your notice is that an atheist who insists that "all powerful" must mean God can do all things whatsoever, and not simply all things logically possible, he's merely quibbling over words. He isn't really objecting to the Christian notion of God; he's only objecting to the term, "all powerful." Maybe we could simply say that, okay, God is not all powerful by that definition. So let's just use a different word so that we can more accurately convey what we mean in regards to God's abilities. Let's say, instead, that God is all mighty which means that God can do all things logically possible. Or we could use any term we want. The important thing is what we mean, and we mean that God can do all things logically possible. Of course there is no incoherence in that.