The a priori and analytic argument posed by Anselm states that God must exist solely by definition; it uses the subject of God and predicate of perfection. In Proslogian 2 Anselm defines God as "being than which no greater can be conceived"  .
Strengths of the Ontological Argument
The main strength of Anselm's argument is showing that the concept of God is not illogical, though explaining that everyone, even a non-believer must have a concept of God in the mind and because of this have a concept of God existing in reality. By definition, God cannot exist in the mind alone, as "no greater can be conceived" as something greater could be thought of, namely that same thing existing in both mind and reality. Hence, God must exist in reality by the meaning of the word God.
The argument succeeds as it is deductive and clear conclusions can be drawn from it, this leaves the argument with only one answer; God exists. Because of it's a priori nature It also offers an actual proof for God's existence which can be logically debated rather than relying on changeable evidence.
Weaknesses of the Ontological Argument
The main weakness of Anselm's argument is posed by Gaunilo of Marmoutier, a contemporary of Anselm, Gaunilo posed, using reductio ad absurdum, that if the logic of the argument were applied to anything other than God, its conclusion would be unreasonable. The analogy of a perfect island was formulated, and using the same argument as Anselm reasoned that as the perfect island could be thought of, then it also must exist.
Though Gainilo's perfect island does have flaws itself as it pointed out by Rowe "if we follow Anselm's reasoning exactly, it does not appear that we can derive an absurdity from the supposition that the island than which none greater is possible does not exist."  It does highlight the many flaws in Anselm's, mainly that something cannot be thought into existence, as Russell states "the argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies."  Russell draws a clear distinction between existence and essence and argues that the essence of something can be defined but this does not constitute existence.
Russell argues that the word "exist" is used incorrectly in Anselm's argument  . Russell states that to define something is to provide an intention and to add existence would be an extension of that intention, Anselm does provide both as the idea which is said to exist, then "that which nothing greater can be conceived" must exist, as it is the totality of all ideas. Russell believes that this only constitutes God being the greatest thing one can think of, but does not prove God exists in realily.
Anselm responds to this with Proslogian 3, by explaining that something that cannot not exist (necessary existence) is greater than something which could not exist (contingent existence), "That God Cannot be Thought Not to Exist"  . This however can be disputed is God is not thought of as a necessity and also as necessary existence can only be applied to God then it is unverifiable.
Another weakness in Anselm's argument is posed by St Aquinas, as Anselm states God is "that which nothing greater can be conceived" then to understand God in this way is to be equal to him, which Anselm cannot be, as he is human.
Descartes begins his argument from the position that he has certain ideas that have necessary qualities; 'necessary'  meaning a priori. For example, it is necessary of a triangle that it has three sides and three angles totalling 180 degrees. Because these properties are demonstrably undeniable, he could not have invented such ideas himself. These ideas possess their own undisputable nature which is, regardless of will, perceived clearly and distinctly. Because of this, they must be true. So, if one is able to draw such an idea from one's mind, then whatever one predicates of that thing is truly a predicate of it. Because it is clearly and distinctly recognised that actual existence is a property of God, actual existence is truly of God's nature. Therefore, Descartes reasons, God exists.
This argument only succeeds if existence is regarded as a predicate; Descartes argues that God possesses necessary existence in the same way that a triangle does three sides; this analogy is a main weakness of Descartes argument. It may be easy to understand what it means for a triangle to have three sides, but if the properties of a triangle, such as the number of its sides, then the mental concept of the triangle changes (and becomes a square/rhombus/etc.) it is very difficult to see how existence, or the lack of it, changes the mental concept of God, and so we may be reluctant to say that it is necessary of God. Kant argues that there are differences between something having certain predicates, such as in the case of the triangle, and something existing, in the case of God.
Kant argues that the examples given by Descartes are of 'judgements'  and it is not necessarily true that three sides exist at all. It is only necessarily true that, given a triangle, there are three sides consisting in it. By this understanding, existence is separate from the predicates that determine how a subject is.
If Kant's view is correct then The Ontological Argument fails, if existence is not a real predicate that is added on to the subject then to deny existence you take away the whole subject. The principle of the Ontological Argument regards this as being as an attribute.
The Ontological argument can only succeed using faith in something which cannot be quantified, therefore can only really be used to prove the existence of God by someone who is already a believer, as Barth suggests "it can tell what theists believe about God but not whether he exists"  . Because of this the argument ultimately fails, as it claims to be a proof but that proof will never be able to be measured.