There are three different types of arguments for the existence of God; cosmological, ontological, and anthropological. I want to here detail them, explain how they are related, and how they are all necessary to give us a fullest apprehension of the concept of God.
Cosmological arguments begin with an analysis of the physical-metaphysical world, and argue to God by determining Him from certain necessities. Cosmological arguments are always inductive; they prove a necessarily existent entity, and then prove the divine attributes of that entity.
Ontological arguments are broader but also more precise, in that it doesn’t consider the world, but only the very idea of God and His relation to being in terms of His existence or non-existence. They are typically reductio ad absurdum; if, for a given definition of God (i.e. the Anselmian “nothing that which can be greater”), if such a being is failed to be instantiated, then it follows that the definition must be of something else. My ontological argument is in the same vein, which moves from our concept of God, this concept’s relation to being, and how being and God cannot fail to be identical.
Anthropological arguments from the tightest cluster of arguments, but can use the most varying of data to prove God from them. They take their data not from the world or being, but from the constitution of man, and consider how if man is such-and-such, and if he could not be such-and-such without such a type of pre-existent being, and so on, there must be a God who instantiated these properties in man to make man man.
Each type of argument is most proficient at a certain element; this is not to say some cannot actually prove God, but only a powerful being; rather, some are more focused on different issues, and the other attributes must be siphoned out of what has just been proved. As an example, Aquinas’ Five Ways (all cosmological, I believe) are only the beginning; there are entire works devoted to proving that this God is omnipotent, omniscient, etc.
Cosmological arguments move from determined entities, and apply logic onto them to see where they point to; as the data set is determined, they are most proficient for proving the impassible attributes of God, i.e. omnipotence, eternality, simplicity.
Ontological arguments move not from entities, but from very ontology itself. They are thus most proficient at understanding and explaining the ontology of God, and setting things in their place in respect to God (from understanding that God is paradigmatically ontological, proto-ontological, or analogously ontological). [I suppose I should devote a post to explaining what I mean by those]
Anthropological arguments move from the very specific entities known as man, hence the anthropos [Greek for 'man'] root. They are dissimilar from cosmological arguments in that they argue from (what are properly whole micro-cosmologies themselves, abiding by separate laws and possessing a different substance) the constitution of man, particularly his intellectual attributes. These arguments take as their data such things as man’s being a subjective entity, his capabilities of reason and moral action, or his non-determined state of being. They are thus most proficient at explaining the relation of man to God, His absolute freedom of will, and omnisciently creative action.
All three types of arguments, if successful for proving God, will by definition then be capable of demonstrating that what is proved is God, so one type of argument can be pursued void of considerations of other types. However, the fullest and most aesthetically appealing picture can only be filled by reference to all three, perhaps even hybrids (say onto-cosmological or onto-anthropological arguments; I’m unsure about the ability to fuse a cosmological-anthropological hybrid, for reasons to explain why). Maybe hybrids are even necessary to link different types together.
There is a certain spectrum here. The ontological argument is primary, because, if God exists, then that should provide sufficient data to prove that He exists, because, abstracting all things contingent down to just what is necessary, we would be left with God, and thus would only have God Himself to argue with. The ontological arguments thus root the cosmological and anthropological arguments. However, there seems to be some irreducibility between the cosmological and anthropological types, for as data sets they are practically irreconcilable without recourse to the ontological dimension of them; thus, there could be no referencing the two to each other without invoking some element of an ontological argument.
Cosmological arguments are certainly easiest to understand and explain, as they move from easily referenced knowledge gained from experience of the world, and is extroverted. Ontological arguments are highly abstract, and so are hard to form, mostly because of how far away removed are its data from us, not from being distant, but from being so near that they are hard to classify by differentiating its data from other elements within its set. Anthropological arguments require a certain understanding of the problems that occur with trying to form a monistic frame (except perhaps a neutral monism, which explains the determinate and mental as species of a more basic substance), and would have a lot to gain from dualism. They are also the most easy to misunderstand and dismiss, since they appeal to introspective data which can only be understood through the intellectual, not physical, senses (i.e. the sensation of qualia understood in opposition to their mere physical data).