The whole argument rests on treating existence as a great-making feature — something a thing is better for having, like power or wisdom. On this view a real hundred-dollar bill is "greater" than an imagined one simply because it exists.
The objection is that existence is not that kind of feature at all. Adding "and it exists" to the idea of a thing tells you the thing is out there; it does not make the idea any richer or the thing any better. As Kant put it, a hundred real dollars contain no more than a hundred possible dollars. And Gaunilo presses it with a joke: by Anselm's logic you could "prove" that the most perfect island must exist, since a real perfect island would be greater than an imagined one — which is absurd.
Defenders answer that the parody quietly changes the subject. Ask what makes an island "perfect" and there is no fixed answer: more palm trees, whiter sand, better weather — for any island you can always imagine a greater one, and "better" depends on your taste. There is no ceiling, so there is no single greatest possible island for the parody to point at in the first place. The being the argument posits is meant to be different in kind. Its greatness is not a matter of degree or preference — more sand, more coconuts — but of perfections like knowledge, power, and goodness that (the defender holds) carry a built-in maximum and are the same for everyone: you cannot have more than complete knowledge, or be better than perfectly good. Anselm's own reply already draws the line — the reasoning applies only to "that than which a greater cannot be conceived," and to nothing else; Alvin Plantinga later put it by saying the greatness of islands has "no intrinsic maximum," while the divine attributes do.
Critics answer that this can look convenient, and that it is not obvious every divine attribute really has a clean maximum — the old puzzle about whether an all-powerful being could make a stone too heavy to lift is meant to press exactly there. And even granting the distinction, one still has to show that such a maximal being is so much as possible.
So the dispute is really two questions at once: whether existence is a perfection at all, and whether the greatness in play is the objective, maxed-out kind the argument needs — infinite knowledge, truth, goodness, the same for everyone — or the loose, taste-relative kind the island trades on. That is where the argument lives.