What is St Anselm of Canterbury's Ontological Argument for the Existence of God?

According to Anselm, God can be defined as 'that than which nothing greater can be conceived'. Everybody possesses such an idea in the mind a priori, an idea of the greatest conceivable being. However, Anslem argues that something that exists in reality as well as in the mind is greater than something that only exists in the mind. Therefore, the idea of the greatest conceivable being cannot be an idea only, for then it would not truly be the greatest conceivable being. As such, the very idea of God demonstrates that God must exist - Anslem cites Psalm 14:1; "The fool says in his heart 'there is no God'". This is an a priori argument, based on logic rather than empirical evidence. This means that if one accepts the premises (a. God is that than which nothing greater can be conceived, b. it is greater to exist in reality than in the mind alone), then one must accept the conclusion (God exists).