This is the strange, bold question of Catholic philosophy. In a certain sense, it is impossible to answer, but by careful application it can nevertheless bear useful fruit. The question can be broken down into two other questions: Does God exist, and Is God a “what,” i.e., Does God have a “whatness”?
The first question is answered by the decision of faith made by a person who is attentive to the words and deeds of Jesus Christ and His Church, and who realizes their implications. The second question is more difficult. God is not a creature; He stands outside of all of physical and spiritual creation. Therefore He cannot fall into any genus. He is not a “type” or a “species”: properly speaking, God is not a “thing” at all.
How does this make any sense? How is it possible for there to be existence without thingness? The ancient Greeks, the fathers of Western philosophy, called this “thingness” essence. Essence is that by which something exists. Through its essence, something is what it is. To put this negatively, if a thing were to lose its essence, it would no longer be itself; it would lack definition; therefore it would lack the wherewithal to exist. Through its daisy-ness a daisy is a daisy. If it had no essence, it could not exist. It would be no-thing.
Now, for the Greeks, essence is always distinct from existence, since the fact that something exists has nothing to do with what it is. For example, nothing about the chair-ness of my seat is making it exist. Chairs do not have to exist just because they are chairs. Chairs exist for some other reason. Of course, someone made the chair, and someone before him made the parts of the chair, and so on; but this does not explain the existence of the chair. It exists, right now, as I am sitting on it. Why? What keeps it there? Certainly nothing about its definition. Therefore, there must exist a being that is not existentially determined, one that does not depend on any other being for its existence. It simply is by necessity.
It is impossible that this Being has an essence distinct from its existence. If it received an act of existence into its essence externally, it would thereby be caused. But this is against its very definition: in order to be the Being from which all beings receive existence, this Being must be Being itself, pure Being existing. Therefore, its essence is existence. Its nature is “to be”.
Notice that “essence” and “existence” are only predicated of this Being in a way that explodes their normal meaning among creatures. Among creatures, existence and essence are always distinct. But we can only conceive of the uncaused Being, depending on nothing for its existence, as the one for which precisely this does not apply. Existence for pure Being is not our creaturely existence, nor is its essence an ontological limit, as ours is. The concept of essence collapsing into existence itself for some uncaused Being is, for Aquinas, the ground of all metaphysics.
Thus, we have answered the second question. God does, in a certain sense, have a “whatness,” but it is identical to His existence. You may ask God, “Do you exist?” and He will say “Yes.” But if you ask Him “What are you? What is your name?” He will answer, as He did to Moses, “I AM.”
In Genesis 1-2, God is revealed as the Creator who is wholly independent of creation. As said above, this means that He belongs to no genus. But God has revealed Himself precisely as the Creator—in fact, this is what Jewish and Christian belief in God signifies. He is separate from the world. He is not created. He is more Real than reality. But since the world receives being from God, God must (I think it is fair to say) be that which confers being, namely, Existence itself existing. I do not think it is untenable to suggest that this is one aspect of the mystery of the Divine Name in Exodus 3 (and Exodus 6). Interestingly, the Divine Name in Hebrew, YHWH (Yahweh), is apparently related to the verb “to be,” YHYH (yihyeh, from hayah).In history this pure Being, the Creator, has revealed Himself as the personal God of the Jews, and through the Jews as the God of and for all people. He reveals Himself as One who is “for” all mankind, One who freely and irrevocably incorporates Himself into mankind even before the Incarnation by making Himself invocable (cf. Ratzinger, Introduction to Christianity). The God of the Old Testament is extremely close: He is never far from those who call on Him. He is closer to His people than they are to themselves. Although this God of ours is above all else personal, the God of Abraham, of Isaac, and of Israel, the God revealed in the Law and the Prophets, nevertheless the basis of His personality is precisely His radical otherness. He is more other than anything in creation because He alone does not belong to the order of creation. He is the Creator; and this is what is signified by “In God, essence is the same as existence.” He does not have a “mode” of being, so to speak, as each creature does. Rather, He is that from which being comes, Being itself. Only in terms of this total otherness does He come to man and reveal Himself as the One for whom all men exist, drawing them to Himself through the Incarnation of Jesus.