"Teleology" is the study of ends... explaining things by their purpose.
"Ontology" is the branch of philosophy dealing with the nature of "being." Closely related to "Metaphysics."
"Cosmology" is the study of WHAT and HOW things (or more generally, the whole universe) exists. "Cosmologists" as a term refers to Pre-Socratic philosophers who were preoccupied with this subject.
"Teleological Proof" tries to prove the existence of God as a "Final Cause" and ultimate goal of the universe. This proof comes from observing the "purposeful" nature of things.
"Ontological Proof" tries to prove the existence of God merely by its being, its ontological definition. That God by definition necessarily exists. This is called an "a priori" proof because it tries to show God just by thinking about God, no need to observe the exterior world.
"Cosmological Proof" tries to prove the existence of God as the creator of the universe, the "original cause" or "unmoved mover." Basically, every things that moves is moved by something, which in turn was moved by something. There must be something at a beginning who had the power to move itself - God. This proof comes from observing the nature of causality in the exterior world.
Which proof you prefer usually coincides with your own philosophy of knowledge and being. Dualists and rationalists, who think knowledge comes from the mind/soul, favor the ontological proof like Augustine, Anselm, and Descartes. Empiricists, who think knowledge comes from observing the exterior world, favor the cosmological proof like Aristotle and Aquinas. If you are a structuralist or determinist you might prefer the teleological.