I will consider this question with particular reference to Descartes' project of the Meditations, and the various proofs that he proposes for God's existence to further his system of knowledge. Descartes is often accused of being circular with respect to God's existence, the classic line being that we have clear and distinct ideas that are guaranteed true by God, and we know God exists because we have a clear and distinct idea of him. However, this is a misrepresentation of Descartes project. He does indeed use his notion of clear and distinct ideas as support for God's existence, but he does not need God as the guarantor of his own existence. Far from being circular, the Meditations does in fact have many lines of argument leading to Descartes feeling justified in saying that God exists, such that if you disagree with one, you may well agree with another. What Descartes does not successfully achieve is a real argument for the nature of God. He ascribes to him `perfection', and suggests what that might mean, but there is no real support for such a line. In order to discover God's naure, we ought perhaps to consider some of the other arguments for his existence.
In the Meditations, there are three principle attempted proofs for God's existence. The first of these is what has become known as the Ontological argument. This argument is that it is inherent in the concept of God that he exists. The argument runs that since our idea of God is that of a perfect being, and existence is a perfection, God must exist. The way Descartes expresses it is that the idea of God must contain the idea of existence, otherwise it is not the idea of God. This seems to mean that it is impossible to assert God's non-existence, or even to question it. This is indeed exactly what St. Anselm, the first proponent of the Ontological argument, says (``it cannot even be thought of as not existing''). This could be used to disallow any argument over whether or not God exists, but most people do not seek to utilise it in that manner. What can be said about the Ontological argument however is that it is far from perfect. Both of its premises are questionable, leaving the structure as reasonable, but empty. The first problem is with our idea of God as a perfect being. You can clearly argue then that if someone doesn't have the idea of God as a perfect being, then, at least for that person, God does not exist. Descartes' response to this is twofold. He begins by saying that you must have an idea of God to be able to talk meaningfully about him, and that if you are to be correct in any assertion you make, then your idea of God must be that of a perfect being, otherwise you are simply referring to the wrong thing, and I assume that he would allow that this thing which is not God could well be supposed not to exist. However, in order to argue meaningfully against God's existence, you must be talking about an idea you have which is of God as a perfect being. Thus, anyone who attempts to claim they have no idea of God, is creating the idea of God in themselves, since otherwise they would not be able to make the claim, being unable to recognise the idea of God.
Another problem with our idea of God is that it is simply that, our idea of him. There seems to be no reason why it is totally necessary that the idea we have is correct. Many of our ideas are wrong, and we are capable of dealing with that, we accept our own fallibility. It seems that we could interpret the first premise as saying that we have an idea of a being who has every perfection, rather than saying there is a being who has every perfection, of whom we have an idea. The shift of emphasis in these two different ways of expressing the first premise change the strength of what we can conclude. If we use the first version, we end up saying that the idea we have cannot admit the idea of non-existence, but says nothing necessarily about any real object. Only the second interpretation allows us to say this being necessarily exists, and at the very least, this exposes an ambiguity. Worse than that, the second mode of interpretation requires us to assume the being's existence before we are able to prove it. This seems to destroy the structure of the argument, and leave us only with the idea that we cannot conceive of his non-existence, rather than any assertion about what is actually true.
This is where Descartes previous reasoning in the Meditations becomes useful. He has already shown that we can never rationally doubt our own existence, as every time we do, we can recognise ourselves as doubting, and thus as existing. He then describes the idea of our own existence as a clear and distinct idea, since it cannot be thought of as true whilst it is being considered, to do so would be incoherent and contradictory. Descartes now uses this theory in his proof of God's existence. He asserts that we have a clear and distinct idea of God, since we cannot logically assert it is a false idea whilst we are considering it, since, as argued before, the idea of God contains the idea of existence. Thus, as a clear and distinct idea, it is self-evidently true, and therefore God exists. What Descartes then goes on to do is use God to underpin the veracity of our clear and distinct ideas about the external world, but these seem to be altogether weaker versions of clarity and distinctness. The first two ideas, one's own existence and God's existence, stand alone without any support required from God.
However, in a return to the Ontological argument, we discover that there are more flaws inherent within it. The idea of existence as a perfection came down from Anselm and his original "being greater than which nothing can be thought." A being as great as can be thought that lacks existence is surpassed by the same being which possesses existence. However, it can be argued that we are making a fundamental flaw when we talk about existence in these terms. The first person to use this line of attack was Kant, who asserted that `exists', whilst functioning as a grammatical predicate, does not in fact function as a logical predicate, and that to therefore speak of something having existence is not in fact to say anything new about that thing. This can be taken further in the notation of Frege and Russell's logic, where it becomes apparent that `exists' behave differently to other predicates. Frege says that this is in fact because it is a second order predicate, one that only functions to ascribe properties to properties. It does not say anything about the object itself. What this means is that we are using `existence' in the wrong way when we speak as though we can give an entity this property. There is no difference in properties between a non-existent object, and the same object that does exist, there is merely an instantiation of such an object. This would mean that a non-existent God is no less perfect than an existent one, and would therefore undermine the ontological argument.
The Ontological argument is not the only way Descartes attempts to prove God's existence. He also attempts to utilise the principle of causal adequacy to formulate what is known as the Maker's Mark argument. The principle of causal adequacy states that every cause contains to an equal or greater extent the properties of its effect. The way Descartes utilises this is by saying that since we have an idea of God, as was discussed earlier, it is beyond our powers to cause such an idea, and that it must therefore have been caused by God. The reason for this is that our idea of God is as an infinite and perfect being, and since we are finite and imperfect, the principle of causal adequacy does not allow for us to be the author of the idea. We can again refer to the earlier argument as to how to deal with those who claim they have no idea of God, and say that they are unable to say that unless they do in fact have such an idea. In fact this idea only requires that one person has the idea of God's perfection, so we could in fact allow that some people do not have the idea of God. However, there is one large potential argument against this. As finite and imperfect beings, if we are unable to be the cause of ideas of perfection and infinity, how are we able to recognise them as truly being ideas of perfection and infinity. We could easily be mistaking near-perfect and near-infinite for the real thing. In that case, we cannot be sure that we could not have been the authors of these imperfect ideas. Descartes response to this is to say that we only need to be able to recognise, not fully comprehend, infinity, and that that is within our powers.
In the Meditations, Descartes offers some convincing support for God's existence, but I would say he falls short of outright proving his existence. As to the nature of God, all we can assert about that is what we believe, not having any evidence of him. The only argument I am aware of that can attempt to ascribe any particular nature to God is the Teleological argument, which I think is too flawed to be of any real use. However, the only possibilities that scrape through the Teleological argument are nothing like the God of Abraham and Isaac. In Hume's Dialogues Concerning Natural Religion it seems that the argument leads us to ideas like a collaborative pantheon, or other possibilities, none of which are the Christian style God. I do not think that there is a proof of God's existence, however neither is there a disproof. Neither side really has any great ground over the other, and quite probably never will.
J. Cottingham, Descartes: Selected Philosophical Writings (New York: CUP, 1999) J. Hick, The Existence of God (New York: Macmillan, 1976) D. Hume, Dialogues Concerning Natural Religion (London: Penguin Books, 1990)