I don’t want to go into a full-dress exposition of the ontological argument, because I think it would be distracting to a simple yet decisive objection to it. For our purposes, then, we can express its structure crudely as follows:
1. It’s possible that there is a necessary being.
2. If it’s possible that there is a necessary being, then a necessary being exists.
3. Therefore, a necessary being exists.
The argument is valid; so, if its premises are true, its conclusion follows of necessity. Well, what reasons can be offered for the premises?
Premise (2) is just an instantiation of Axiom S5 of S5 modal logic. The underlying idea of Axiom S5 is that what is necesssarily the case doesn't vary from possible world to possible world: if something is necessary in one possible world, it's necessary in every possible world. I accept Axiom S5; so I accept premise (2). That leaves us with premise (1). Is it more reasonable to believe it than not -- or at least: is it more reasonable to believe it than to suspend judgment either way?
No, it isn’t. For the evidence is supposed to be that it’s conceivable that such a being exists, and that whatever is conceivable is possible. Now there are a lot of points that could be brought up here, but I want to limit myself to one point based on recent work in modal epistemology, i.e., the study of how our beliefs about what is impossible, possible, and necessary are known and/or justified.
There are many objections, both classical and contemporary, that have been raised against inferences from conceivability to possibility. For example, in the past, people were able to conceive of the Morning Star existing without the Evening Star, or water existing without H20. So if everything conceivable were possible, it should follow that it’s possible for the Morning Star to exist without the Evening Star, or water without H20. But we now know that these things are impossible, since the Morning Star is the Evening Star, and water is H20.
Another example: Goldbach's Conjecture is the mathematical hypothesis that every even number greater than 2 is the sum of two primes. To date, no mathematician has proven that Goldbach's Conjecture is true (nor have they proven that it's false). Now I can conceive, in some sense, that Goldbach's Conjecture is false. I can also imagine that it's true. So if all inferences from conceivability to possibility are valid, then it follows that it's both possible for Goldbach's Conjecture to be true, and possible for Goldbach's Conjecture to be false -- in other words it would follow that Goldbach's Conjecture is only contingently true if true at all. But that can't be right, for mathematical statements are necessarily true or necessarily false if true or false at all!
Thus, it looks as though we need some criterion of legitimate conceivings to screen out illegitimate conceivings, thereby preserving the utility of inferences from conceivability to possibility.
A lot of progress has been made over the past several decades in the sub-field of modal epistemology, but for our purposes, it’s enough to mention one key distinction that’s been developed that’s helpful. Stephen Yablo and James Van Cleve have each pointed out that there’s a distinction between not conceiving that P is impossible, on the one hand, and conceiving that P is possible, on the other. Van Cleve calls the former, ‘weak conceivability’, and the latter, ‘strong conceivability’.
Now it turns out that pretty much all of the counterexamples to the conceivability-possibility inference are cases in which something is weakly conceivable. For example, when one says that they can conceive of Goldbach’s Conjecture being true, and that they can conceive of it also being false, they really mean that they can’t see that either conception is impossible – i.e., they only weakly conceive of such things. The same goes for conceiving of water existing without H20, and conceiving of the Morning Star existing without the Evening Star. By contrast, I can strongly conceiving of my car as being red, and of myself as a person who doesn't like to surf (albeit just barely!); thus such conceivings provide prima facie evidence that it's possible for my car to be red, and that I really could have been a person who doesn't enjoy surfing.
In light of this distinction, then, we can handle the counterexamples by limiting conceivability-possibility inferences to those that involve what is strongly conceivable – i.e., to those in which one intuits that p is possible, and not to those in which one merely fails to intuit that p is impossible.
With the weak/strong conceivability distinction before us, let’s consider premise (1) again. Is it strongly conceivable that there is a necessary being -- i.e., do we "just see" that it is possible? It doesn’t seem so. Rather it merely seems weakly conceivable – i.e. I merely can't intuit that such a being is impossible. But this isn’t enough to justify the key premise (1) of the ontological argument. For that to be so, a necessarily existing individual would have to be strongly conceivable.
To come at the point from another direction: Christian theistic philosopher Peter Van Inwagen asks us to imagine a being whom he calls 'Know-No'. Know-No is a being who knows that there are no necessary beings. If such a being is possible, then a necessary being is impossible. For then there would be a possible world in which a being knows that there is no necessary being. And if he or she knows it, then it's true that there's no necessary being.
Now both possibilities can't be true -- either a necessary being is possible, or a being like Know-No is possible, but not both, since the possibility of each one precludes the possibiliity of the other. But notice: both possibilities are conceivable in the weak sense: on reflection, I fail to see an incoherence in the conception of either one. So, if weak conceivability were sufficient evidence for possibility, it would follow that I'm justified in believing that necessary beings and Know-Nos are both possible, which, as we've just seen, is false -- if either one is possible, the other is impossible. Thus, again, the notion of a necessarily existent individual is only weakly conceivable, and weak conceivability isn't good evidence for possibility.
Thus, it looks as though the ontological argument is not a successful piece of natural theology. Whether or not the key premise is true, I don’t have sufficient reason to think so. Thus, the argument is of no help in the task of justifying theism.
 “Is Conceivability a Guide to Possibility?”, Philosophy and Phenomenological Research 53 (1993), 1-42.
 “Conceivability and the Cartesian Argument for Dualism”, Pacific Philosophical Quarterly 64, (1983), 35-45.
 This objection to the ontological argument can be found in Peter Van Inwagen's textbook, Metaphysics, 2nd edition (Westview, 2002).