The following video presents a good explanation of the ontological argument. It also allows us to examine the distinction between analytic philosophy, which the ontological argument is part of, and realist philosophy, which could be represented by the thought of Thomas Aquinas. First, watch the video.
Note that this video does a good job of showing the shallowness of the typical response to the ontological argument. The response has not changed much from the first attempt to refute it. A thinker named Gaunilo proposed the existence of a perfect island; if I can conceive of a perfect island, it must exist. But as this video shows, finite things like islands and unicorns are not what the ontological argument is dealing with.
But more importantly, note the explanation of possible worlds. They are defined as hypothetical situations. In this case analytic philosophy starts with all hypothetical situations (all possible worlds) and argues to existence from that point. The distinction between this perspective and realist arguments is that the realist starts with what is known about reality. The person starting from an analytic position is locked in a thought problem where the goal is to get to the real world. By contrast, the realist starts from something known in reality and argues to conclusions about both reality and concepts. A Christian realist, such as Aquinas, starts with two things: revelation from God and what we know about reality. He then attempts to align both of these with each other and with concepts in the mind. By starting with God’s revelation plus knowledge of reality, then attempting to draw conclusions, Aquinas rigorously evaluated all that he found around him. Thomists have done so ever since, and we believe that the modified realist perspective is the most sound, and fits best with what is known in all fields of modern thought and theology.
I have found that many, if not most, philosophy students today are taught to start from an analytic perspective. They are taught analytic philosophy so much that that they know of no other perspective…they are like fish who never question that the world is made of water, until one day they are brought into the air and find it incomprehensible. Analytic philosophers are not taught realism, or if they are, they are merely taught that we humans cannot trust our sense perceptions enough to prove there is a real world out there. The realist responds with a simple answer: we must start with 1) there is a real world outside of my mind, and 2) I can know something about it. If we do not start with these premises, we will never prove them using philosophy. One philosopher, Rene Descartes, proved to us that if we start within the mind, we are forever stuck there, making futile attempts to get to reality.
I have taken to using the following response to my analytic friends: If a possible horse fell into a possible mud puddle, nothing would actually have to get washed off. As long as we are dealing in hypotheticals, we will always have trouble getting the conclusions back into reality. This is why Thomist realists generally do not use the ontological argument. The reason we usually do not use it is not because we think it is unsound, but because it does not begin with known observable facts in the universe or with revelation from God. Those trained in the analytic tradition are so steeped in hypothetical thought problems that they often do not question the field of play that they have chosen for themselves, as the video above demonstrates.
For more information on the distinction between analytic and realist thought, see The Two Logics by Henry Veatch.
So what are we to make of the ontological argument? While I am not of the caliber of philosopher to claim to deal with all issues around this argument, I nevertheless make my humble response. In a sentence, the argument is not refutable, but I do not think is as strong as its supporters make it. By this I mean that the argument is logically solid and as stated it cannot be refuted. But since it is at its root a thought problem, then its conclusions are thought conclusions. The argument is hanging in mid-air begging to touch down onto reality somewhere, but we never seem to be able to tell whether it has done so. Or if it does, it does so in an awfully slippery manner.