Question: "What is the Teleological argument for the existence of God?"
Answer: The word teleology comes from telos, which means "purpose" or "goal." The idea is that it takes a "purposer" to have purpose, and so, where we see things obviously intended for a purpose, we can assume that those things were made for a reason. In other words, a design implies a designer. We instinctively make these connections all the time. The difference between the Grand Canyon and Mount Rushmore is obvious—one is designed, one is not. The Grand Canyon was clearly formed by non-rational, natural processes, whereas Mount Rushmore was clearly created by an intelligent being—a designer. When we are walking on a beach and find a wristwatch, we do not assume that time and random chance produced the watch from blowing sand. Why? Because it has the clear marks of design—it has a purpose, it conveys information, it is specifically complex, etc. In no scientific field is design considered to be spontaneous; it always implies a designer, and the greater the design, the greater the designer. Thus, taking the assumptions of science, the universe would require a designer beyond itself (i.e., a supernatural designer).
The teleological argument applies this principle to the whole universe. If designs imply a designer, and the universe shows marks of design, then the universe was designed. Clearly, every life form in Earth's history has been highly complex. A single strand of DNA equates to one volume of the Encyclopedia Britannica. The human brain has approximately 10 billion gigabytes of capacity. Besides living things here on Earth, the whole universe seems designed for life. Literally hundreds of conditions are required for life on Earth—everything from the mass density of the universe down to earthquake activity must be fine-tuned in order for life to survive. The random chance of all these things occurring is literally beyond imagination. The odds are many orders of magnitude higher than the number of atomic particles in the whole universe! With this much design, it is difficult to believe that we are simply an accident. In fact, top atheist/philosopher Antony Flew's recent conversion to theism was based largely on this argument.
In addition to demonstrating God's existence, the teleological argument exposes shortcomings in the theory of evolution. The Intelligent Design movement in science applies information theory to life systems and shows that chance cannot even begin to explain life’s complexity. In fact, even single-celled bacteria are so complex that, without all of their parts working together at the same time, they would have no survival potential. That means those parts could not have developed by chance. Darwin recognized that this might be a problem someday just by looking at the human eye. Little did he know that even single-celled creatures have too much complexity to explain without a creator!