God. Real. Right.

In one of his famous letters to John Adams, Thomas Jefferson makes the case that when arguing for the existence of God the teleological argument is better than the cosmological argument. His reasoning is straightforward and dependent on the rise of science’s authority in the world of his day.
The cosmological argument traditionally claims that the universe must be caused by something, and the thing that causes it is God. (A very brief but better form of the argument is here.) But Jefferson objects with a foundational principle of the scientific worldview, the law of parsimony (Occam’s Razor). If a scientifically minded person must choose between a temporal universe with an eternal God on one hand, and an eternal universe by itself on the other hand, the scientist (by the law of parsimony) will choose the universe by itself. So Jefferson reasons that the many discoveries of his day regarding the complexity and intricacy of nature–particularly biological nature–are a better source for arguments in favor of God’s existence. With that approach (the teleological argument), Jefferson manages to respect science and argue for the existence of an intelligent designer.
But he unnecessarily sacrifices some things too. There is a real disadvantage to relying on the teleological argument. Particularly, there is what is known as the “God of the gaps.” That is, when people cannot explain something and they credit it to God solely on that basis, the concept of “God” is simply what they are using to fill in the “gaps” in their knowledge. But, of course, as knowledge grows, the gaps are filled and God is pushed back further and further in their minds. Ultimately, this approach to the teleological argument leads to Deism–the view that even if God did create this finely tuned system, there is no reason for Him to be involved in it now. Deism is obviously antithetical to anything like biblical Christianity. But it was a popular view of the world in and just after Jefferson’s day. It can also lead (albeit errantly) to naturalism and atheism as scientists, who are inductive thinkers by practice, inductively reason that since they have filled so many gaps in knowledge already, it is inevitable that they will eventually fill the rest, ruling out the existence of God. Yes, it is very poor reasoning, even inductively, but it is common.
Enter William Paley. He publishes a much more sophisticated form of the teleological argument in his Natural Theology in 1802. The core of the argument is the meaning of design:

…when we come to inspect the watch, we perceive (what we could not discover in the stone) that its several parts are framed and put together for a purpose, e. g., that they are so formed and adjusted as to produce motion, and that motion so regulated as to point out the hour of the day; that, if the different parts had been differently shaped from what they are, of a different size from what they are, or placed after any other manner, or in any other order, than that in which they are placed, either no motion at all would have been carried on in the machine, or none which would have answered the use that is now served by it.

The point is not simply about complexity. It is about a disparity between the nature of the materials and their form (or, in this case, function). That is, there must be an intelligent designer whenever nothing intrinsic to the material would have produced the form it has taken or function it is serving. And so the teleological argument for the existence of God flourished for more than sixty years as biology revealed ever-increasing levels of form and function completely inexplicable in terms of its material substance. How could an eye just happen to see, after all!
Then comes evolutionary theory, in which the basic claim is that things that tend to replicate themselves just do tend to replicate themselves–that is, they reproduce. And in a competitive environment (wherever there are limited resources; aka everywhere), the best reproducers continue to do so even when less effective competitors fail–that is, they survive. So in one book, Darwin’s theory bridges the supposed “gap” in Paley’s explanation, and pushes God back a step. To be clear, Darwin attempts to make the argument that the material nature of biological organisms does account for their form and function. Darwin acknowledges, though, that his theory does not eliminate the Creator in his Origin of Species:

Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved.

His acknowledgment of the Creator is almost gracious. But not really. It is at best simply evidence of the Deism inherent in evolution (especially in that day). “God can be given credit for creating the law, but it is the law which governs life now.”
But as with all “God of the gaps” dialogs, the next step is inevitable. A dozen years after Origin, Darwin argues in The Descent of Man not simply for the evolution of animal species, but also for the case that the evolutionary path crosses what had previously been a sacred boundary, the division between animal and human.

For my own part I would as soon be descended from that heroic little monkey, who braved his dreaded enemy in order to save the life of his keeper, or from that old baboon, who descending from the mountains, carried away in triumph his young comrade from a crowd of astonished dogs–as from a savage who delights to torture his enemies, offers up bloody sacrifices, practices infanticide without remorse, treats his wives like slaves, knows no decency, and is haunted by the grossest superstitions.
Man may be excused for feeling some pride at having risen, though not through his own exertions, to the very summit of the organic scale; and the fact of his having thus risen, instead of having been aboriginally placed there, may give him hope for a still higher destiny in the distant future. But we are not here concerned with hopes or fears, only with the truth as far as our reason permits us to discover it; and I have given the evidence to the best of my ability. We must, however, acknowledge, as it seems to me, that man with all his noble qualities, with sympathy which feels for the most debased, with benevolence which extends not only to other men but to the humblest living creature, with his god-like intellect which has penetrated into the movements and constitution of the solar system–with all these exalted powers–Man still bears in his bodily frame the indelible stamp of his lowly origin.

Darwin is very carefully closing that book with the case that he would rather be related to moral primates than superstitious aboriginal humans. He is fully committed by then to leaving behind belief in God in favor of a morality defined by survival. (The scary nature of that kind of morality is a topic for a different day, from Schopenhauer to Nietzsche and beyond.)
As evolution has grown into the governing paradigm of the scientific community, so has grown the rejection of intelligent design from scientific discourse. No surprise there.
So what is the moral of this story for Christian apologists? First, appealing to what science has not figured out yet is not the best basis for an apologetic (an argument in favor of God). Second, allowing science to define the playing field is bad apologetics, not because science will win some strange argument against God–nothing could be further from the truth–but because by its very nature science can only study matter, and cannot study the things that matter, for instance, the being or nature of God. (More for another day!)
And so, while Intelligent Design arguments and cases like the Anthropic Principle are fascinating, interesting, and practically appealing or maybe even compelling for the moment, they are not ultimately the soundest arguments for God’s existence.
For a smattering of arguments (to be expanded) less contingent on such a tendentiously naturalistic discipline, see the page on God.