This particular argument is not greatly popular nor used very often today, but it is one of the classic 'proofs' for the existence of God and so deserves a mention. (Besides, I quite like it)
The ontological argument was first formulated by Anselm of Canterbury. He began by defining God as "a being than which nothing greater can be conceived". Anselm argued that there is a crucial difference between a greatest concievable being which only exists in the mind, and a greatest conceivable being which exists in reality. Anselm argued that if the most perfect conceivable being only existed in the mind, we would have the problem that we could conceive of a greater being, namely one which exists in reality.
"If then that-than-which-a-greater-cannot-be-thought exists in the mind alone, this same that-than-which-a-greater-cannot-be-thought is that-than-which-a-greater-can-be-thought. But this is obviously impossible. Therefore there is absolutely no doubt that something-than-which-a-greater-can-not-be-thought exists both in the mind and in reality." (Anselm, Proslogion Chapter 2 in "St Anselm's Proslogion" M J Charlesworth, Oxford Clarendon Press, 1965)
Later on in Anselm's book, he produced what became known as the second form of the argument. If God is the most perfect being there is, unlimited in time, then it is impossible for him ever to have come to exist or ever to cease to exist - he has aseity, self-existence. This means he necessarily exists, and needs nothing in order to exist.
"For something can be thought that cannot be thought not to exist. Hence, if that-than-which-a-greater-cannot-be-thought can be thought not to exist, then that-than-which-a-greater-cannot-be-thought is not the same as that-than-which-a-greater-cannot-be-thought, which is absurd. Something-than-which-a-greater-cannot-be-thought exists so truly then, that it cannot be even thought not to exist." (Anselm, ibid Ch3)
Anselm referred to the fool who says in his heart 'there is no God' (Ps 14:1, 53:1) - even such a person has the idea of God as the most perfect conceivable being, and such a being must exist. This is similar (though rather more intelligent) to the arguments of some modern-day Christians, who insist that atheists know God exists, they just pretend not to. (See www.tencommandments.org R T Lee's site repeats some of these type of arguments.) Anselm's first critic, Gaunilon, wrote a book entitled "On Behalf of the Fool". He said that Anselm's argument, if applied to other areas than the existence of God, leads to absurdity. For example, if we think of the greatest conceivable island, according to Anselm, such an island must exist, otherwise it cannot be the most perfect conceivable island! Anselm replied to Gaunilon, in effect stating that God is a special case. It is only he who has aseity, and is not contingent upon anything else for his existence.
Rene Descartes (1596-1650) reformulated the ontological argument, and drew a great deal of attention to it by so doing. Descartes argued that existence is one of the defining predicates of God. As the fact that its internal angles are equal to right angles is a necessary characteristic or predicate of a triangle, so existence is a necessary predicate of a supreme being. As a triangle without its characteristics would not be a triangle, so God without existence would not be God.
This idea was challenged by Immanuel Kant (1724-1804). He accepted (to a point) that existence is a predicate of a supreme being, just as the idea of three angles is a predicate of a triangle. But both of these are only true if there is a triangle, and if there is an infinitely perfect being.
"To posit a triangle, and yet to reject its three angles, is self-contradictory; but there is no self-contradiction in rejecting the triangle altogether with its three angles. The same holds true of the concept of an absolutely necessary being." (Immanuel Kant, "Critique of Pure Reason" trans N. Kemp Smith, Macmillan & Company Ltd, 1933 "Transcendental Dialectic", Book II, Chapter 3, Section 4)
Kant also rejected the assumption that existence is a predicate that something can either have or lack. He pointed out that the idea of existence does not add anything to the concept of a thing. For example, an imaginary $100 has the same number of dollars in it, as a real $100. If we affirm that it exists, we are only applying the concept of $100 to the world.
So all a definition of God does is to describe your idea of God, it cannot prove such a being exists. It is worthwhile to note, however, that Anselm's argument was not made originally for the purpose of proving God's existence to non-believers, but to aid those who already had faith. For this reason, the ontological argument is worth keeping for those who have, as Anselm put it, "faith seeking understanding".