More in-depth of Anselm

Anselm's argument is an a Priori argument, meaning it comes from reason alone. And That than which nothing greater can be thought simply means the best thing possible to think of. 

Anselm is addressing his argument to Atheists and calls them Fools, as they are being irrational saying God does not exist.

The contradiction msde in section 4 is that the fool is thinking what cannot be thought. He understands the concept of God existing in the mind and in reality but says that God only exists in the mind, they deny God's existence in reality and the mind even after saying they understand that he exists in the mind and in reality. 

Gaunilo's criticism says that just because you define something as being perfect doesn't mean it exists. Anselm's argument is used to prove the existence of any perfect thing such as a perfect island. Any perfect thing has to exist, is what Gaunilo is trying to get at. 

The concept in Proslogian 3 that is introduced is that - God is of necessary existence. 

This concept strengthens Anselm's argument as it means his argument can only work for God as everything else is contingent.

Just because you define something as perfect does not mean it will exist. 

The Perfect Island
That Island than which no greater can be thought.

 An island that exists in the mind only is not as great as an island that exists in both the mind and reality. 

Therefore by definition that island than which no greater can be thought must exist in both the mind and in reality.