Ah, good ol Anselm's ontological argument. I think it's very clever. However I will now expose a fallacy in this version. (There are better versions out there, such as Plantinga's Modal version)
let me outline the argument:
definition: God is that being than which no greater can be concieved
(1) God exists in the understanding but not reality.
(2) A being otherwise like God but existing also in reality is conceivable [premise]
(3) Existence in reality and in the understanding is greater than existence in the understanding alone. [premise]
(4) A being greater than God is conceivable [(1)(2)(3)]
(5) It is possible to concieve of a being greater than that Being greater than which cannot be conceived [(4), definition]
But (5) is a contradiction, therefore, assumption (1) is false, But:
(6) God exists in the understanding [premise]
(7) God exists in reality
Here's the fallacy in Anselm's argument
step (5) might be read in two different ways. i.e. it is ambiguous
(5') There exists a being which is (actually) such that no greater being can possibly be conceived and it is possible to conceive of a greater being than that.
(5'') It is possible to conceive the following: Something is (actually) greater than that being greater than which cannot be conceived [which latter may be merely possible]
Anselm's principle: If something is defined to be "the F" then a possible thing satisfying the definition is actually an F.
But this is principle is not true.
That's one way of going about things. However, probably the most famous objection to the ontological argument was put forth by Kant. He argues that 'existence' isn't a predicate (property) of things. This is still being debated today.
And I want to note one last thing. See Dr. Why's answer? That is an example of a parody of the ontological argument. There are tons out there, and they are all meant to show that there is something wrong with the argument, because you can prove the existence of a plethora of things using parallel reasoning. The most famous of these parodies was put forth by the monk Ganuilo, and it is commonly referred to as the 'Perfect Island objection'. Using the same reasoning Ganulio tries to prove the existence of the most perfect island.
However, this objection is flawed. And it has to do with what exactly defines the greatest possible island? perhaps number of palm trees? dancing hula girls on the beach? a thousand maybe? well these are all characteristics with no intrinsic maximums. Meaning that if you define the perfect island as having a thousand hula dancers, you can also add one more. would that not be an even greater island? The conslusion being that the analogy is false, because islands don't have intrinsic maximums, however, God does. (the maximally great properties for God is being all-powerful, all good, and all knowing. These have maximal caps.)
However, the objector can accept this fact, and reformulate an objection that, perhaps doesn't prove the GREATEST possible being, but proves a bunch of almost perfect beings, such as demi-gods. This is a BAD conclusion for the monotheist. (if you're a polytheist I suppose its okay)
Nice question btw. The ontological argument has a history and some big names in philosophy have made versions of it. Famous logician and mathematician Gottlob Frege has a version. Alvin Plantinga has a modal version. Rene Descartes accepts it, and has a version. And of course Anselm himself (which is the classic version that you stated)