Alex Byrne wrote a pretty good article in the Boston Review going over the two kinds of popular arguments for the existence of God: the ontological and the teleological arguments. Byrne’s article gives you the basic overview of these kind of arguments, although if you want more technical details, you can read their entries in the Stanford Encyclopedia of Philosophy.
After providing an overview of the ontological and the teleological arguments for the existence of God, Byrne concludes, unsurprisingly, that neither argument offers persuasive evidence of God’s existence. However, I really liked Byrne’s way of putting the conclusion in the broader context of belief:
“If a persuasive argument for the existence of God is wanted, then philosophy has come up empty. The traditional arguments have much to teach us, but concentrating on them can disguise a simple but important point. As Anselm and Paley both recognized, the devout are not exactly holding their collective breath. For the most part, they do not believe that God exists on the basis of any argument. How they know that God exists, if they do, is itself unknown—the devout do not know that God exists in the way it is known that dinosaurs existed, or that there exist infinitely many prime numbers. The funny thing about arguments for the existence of God is that, if they succeed, they were never needed in the first place. “
In drawing this conclusion, Byrne shifts the discussion from the metaphysical/ontological (whether God exists and what properties he has) to the epistemological: that is, how do believers know that their God exists? If, as Byrne argues, that believers’ epistemology do not rest on logical arguments, then the whole enterprise of proving God’s existence through logic is doomed to futility.
Note, however, that Byrne’s conclusion is not that no one can prove that God does not exist: his conclusion is that God’s existence can’t be proven using logical methods of argument. It’s also important to be aware of the scope of the argument: the refutation of the ontological and teleological arguments only prove that God, as characterized by Abrahamic religions, does not exist. Refuting the ontological/teleological argument does not entail the non-existence of non-Abrahamic deities.