Answer
Natural theology is the study of God based on the observation of nature, as distinct from "supernatural" or revealed theology, which is based on special revelation. Because observing nature is an intellectual pursuit, natural theology involves human philosophy and reasoning as means of knowing God.
By examining the structure and function of a snapdragon bloom, I might reasonably conclude that the God who created the snapdragon is powerful and wise—that is natural theology. By examining the context and meaning of John 3:16, I might reasonably conclude that God is loving and generous—that is revealed theology.
The division of theology into "natural" and "revealed" had its roots in the writings of Catholic theologian Thomas Aquinas (A.D. 1224 - 1274). In an attempt to apply Aristotelian logic to the Christian faith, Aquinas emphasized man’s ability to comprehend certain truths about God from nature alone. However, Aquinas maintained that human reason was still secondary to God’s revelation, as taught by the Church. Aquinas was careful to distinguish what could be learned through "natural reason" from doctrinal tenets, calling the truths gleaned from nature "preambles to the articles [of faith]" (Summa Theologica, First Part, Question 2, Article 2). That is, reason may lead to faith, but it cannot replace faith.
Later theologians took Aquinas’s idea and expanded it. Other writers emphasizing natural theology were Samuel Clarke, William Paley, and Immanuel Kant. Over the years, the miraculous was downplayed as Christianity was reduced more and more to a "rational" philosophy.
The deists relied solely on natural theology for their knowledge of God, to the complete exclusion of special revelation. To the deist, God is unknowable except through nature, and the Bible is unnecessary. This is why Thomas Jefferson, a deist, literally cut all the accounts of miracles from his Bible—Jefferson wanted a natural theology only.
The Romantic poets, as a whole, were proponents of natural theology. Although they stressed man’s emotion over his intellect, they were constantly extolling the virtue and transcendence of nature. One very clear presentation of natural theology is William Wordsworth’s famous poem "The Rainbow," which ends with these lines: "And I could wish my days to be / Bound each to each by natural piety." Wordsworth expressly wishes for a "natural" (versus a "supernatural") piety. His spirituality is rooted in the natural world; the joy he feels at the sight of a rainbow is, for him, the truest worship of God. Those today who say, "I feel closer to God on a walk through the forest than I do at church" are expressing Wordsworth’s brand of natural theology.
An undue emphasis on natural theology has even accommodated pantheism. Some have gone past the idea that nature is an expression of God to the idea that nature is an extension of God. Since, the logic goes, we are part of nature, then we are all a little part of God, and we can therefore know Him.
In more modern times, "natural theology" can also refer to the attempt to synthesize human knowledge from every area of science, religion, history, and the arts. The new natural theology pursues a transcendent "encompassing reality" in which mankind exists, but the focus is humanity, not God; consequently, it is really another form of humanism.
Here are some biblical points concerning natural theology:
1) The Bible teaches that a basic understanding of God can be gained from the natural world; specifically, we can see "his eternal power and divine nature" (Romans 1:20). We call this "general revelation" (see also Psalm 19:1-3).
2) The context of Romans 1 indicates that such a basic understanding of God’s existence and power is not enough to lead a person to salvation. In fact, the pagan’s inherent knowledge of God (through nature) has been distorted, leading to judgment rather than to salvation.
3) Natural theology can cause someone to theorize that God is invisible, omnipotent, and wise, but these are all abstract characteristics of an unnamed "Supreme Being." Natural theology cannot teach the love, mercy, or judgment of God, and it is worthless for bringing anyone to saving faith in Jesus Christ. "How can they hear without someone preaching to them?" (Romans 10:14).
4) The fall of man has affected the whole person, including the intellect. A reliance on natural theology assumes that human reason has not been tainted by original sin, yet Scripture speaks of the "depraved mind" (Romans 1:28), the "sinful mind" (Romans 8:7), the "corrupt mind" (1 Timothy 6:5), the "dull" mind (2 Corinthians 3:14), the "blinded" mind (2 Corinthians 4:4), and the need for the mind to be renewed (Romans 12:2).
Natural theology is useful insofar as God has created the world and the world still points to Him as Creator. However, given the fallen state of our intellect, we cannot properly interpret even that without God’s special revelation. We need God’s gracious intervention to find our way back to Him. What we need more than anything is faith in the Bible and in Jesus Christ (2 Peter 1:19).