"What is the Ontological argument for the existence of God?"
Answer: The ontological argument is an argument based not on observation of the world (like the cosmological and teleological arguments) but rather from reason alone. Specifically, the ontological argument reasons from the study of being (ontology). The first and most popular form goes back to St. Anselm in the 11th century A.D. He begins with stating that the concept of God is "a being than which no greater can be conceived." Since existence is possible, and to exist is greater than to not exist, then God must exist (if God did not exist, then a greater being could be conceived, but that is self defeating—you can't have something greater than that which no greater can be conceived!). Therefore, God must exist. Descartes did much the same thing only starting from the idea of a perfect being.
Atheist extraordinaire Bertrand Russell said that it is much easier to say that the ontological argument is no good than it is to say exactly what is wrong with it! However, ontological arguments are not terribly popular in most Christian circles these days. First, they seem to beg the question as to what God is like. Second, subjective appeal is low for non-believers as these arguments tend to lack hard objective support. Third, it is difficult to simply state that something must exist by definition. Without good philosophical support for why a thing must exist, simply defining something into existence is not good philosophy (like stating that unicorns are magical, single-horned horses that exist). These problems notwithstanding, several prominent philosophers today continue to work on this more unusual form of theological argument.