It tends to get tossed around quite frequently in conversations of this variety that "You can't prove a negative." But this is only half true.

Take for example, an apple that is an orange. This sounds crazy, and intuitively, we know this object doesn't exist. Why, you might ask? Because its existence contradicts itself. If it is an apple, it is not an orange, and vice versa; we have an internal inconsistency that is immediately apparent. To say it explicitly: one of the properties of being an orange could be called not-appleness, and one of the properties of being an apple could be called appleness. Appleness and not-appleness being properties of the same object is a contradiction, therefore the existence of this object is impossible, not just improbable; as in its existence would destroy the universe as we know it in an incredibly uncool explosion of logic.

So what does this have to do with Atheism? Well, let's say we come across a particularly amazing philosopher. If he can clearly define what we mean by "God" (thank you Cletus) and definitively show that it is internally inconsistent (anyone care to try?), than he has disproven the existence of said "God."

The historical problem is that most all proofs of the Judeo-Christian God's non-existence involve a contradiction between arbitrary properties of said deity and some principle or observation of the universe. Therefore, the observation can either be explained away (the Problem of Evil) or the deity's nature redefined (Evolution (as a tool (to trick us (for fun)))). The tremendous task of conjuring a comprehensive definition of "God" and demonstrating internal inconsistency would require true genius.