The Transcendental Argument is a fairly easy argument to misunderstand. In a nut shell, it states that because there are things which observably exist which transcend matter/energy, that therefore atheism must be false.
It is important to note that like the morality argument mentioned in an earlier post, the transcendental argument is polemical rather than apologetic. By this, I mean that it is criticizing atheism, rather than defending Christianity or theism. However, unlike the morality argument, it is a logical proof, rather than an appeal to what is best for society. It does not seek to say that Christianity is merely socially superior to atheism, but to show that atheism is philosophically untenable.
Therefore, it is inaccurate to say that this argument proves that God exists. It does not demonstrate monotheism, nor does it demonstrate a sentient transcendent being, though I will argue later that it does imply it. It is more accurate to say that this argument disproves materialism, which is a necessary tenant of atheism itself.
Materialism: True Or False
Materialism is the philosophic belief that only matter-energy exists. Everything else is imaginary, illusionary, or conventional. Atheism believes in Materialism as a central tenant.
The logic to disprove this flows as follows:
P1: Materialism is true if and only if everything which exists is matter-energy
P2: 'A' is not matter-energy
P3: 'A' exists
C1: Therefore, there exists something which is not matter-energy
C2: Therefore Materialism is false.
The trick of course is to find something which satisfies 'A'. We call this a transcendental: something which transcends physical reality. The debate usually is over premise 3, and what constitutes existence. If 'A' can be shown to be possibly conventional or illusionary, then it cannot be said that 'A' satisfies the argument. Theists have proposed several transcendentals which we feel satisfy the argument.
Examples of Transcendentals
Unfortunately, the existance of a transcendent universal moral code is probably the most common transcendental proposed. Now, I say unfortunately for a few reasons.
First of all, it causes people to confuse the Transcendental Argument with the Morality Argument. This is often done by those that the argument is aimed towards, and by those attempting to use the argument. I think both arguments suffer for this, since it makes the Transcendantal Argument appear less formal, and the Morality Argument to simply be muddled.
Second of all, it is incredibly difficult to demonstrate that such a moral code objectively exists. Atheists often can easily rhetorically avoid this by claiming that morality is conventional in nature.
Third, I'm not even sure if I would say that such a moral code exists. While I believe there to be one law by which we will all be judged, I consider this to be administered by a divine government, not something which can be said to exist in and of itself. I have yet to find an argument which truly demonstrates that a certain moral code is woven into the fabric of reality, or how such a code can be demonstrated.
The original transcendental proposed was math and logic, and I think there is something to this. There is some debate about whether logic is derivative of math, or whether they simply can overlap, but I don't really think it matters for the purpose of this argument.
So how can we say that math/logic exists, and are not merely conventions? Simply put, because they are universal. The rules of math/logic are not invented, but rather discovered, and are the same in all societies, though they may be represented differently.
Since it is not a convention, atheists must try to say that that it is illusionary. However, nothing illusionary can have an effect on something real, and it appears that math/logic affects reality. Proving it is a bit more difficult, but I believe it to be doable.
A better transcendental, IMO, is one that I proposed a couple of years ago: thoughts. This is more than an appeal to Descartes. I'm not talking about self-evident subjective experience of thinking. Thoughts clearly exist in some sense, but what must be demonstrated is that they exist distinctly, and are not simply illusions of the brain.
No one really proposes that thoughts are conventions. They are not sociological constructs, but individual manifestations of the subjective mind. However, thoughts produce sociological constructs, i.e. ideas. And ideas change reality.
Something which is illusionary cannot affect that which is real. Yet, ideas cause racial separations, the impractical weaving of material (i.e. decorations), and the radical alterations of physical objects (launching satellites into space, and detonating atomic bombs). It seems to me that explaining such occurrences in purely physical terms is unparsimonious, if not down right impossible.
The last transcendental we'll deal with here is information. It can be easily demonstrated that information is greater than the medium within which it is represented. Clearly language is conventional in nature, and cannot be said to properly exist, but the fact that symbols can be carved into a rock, which allows me to unearth an object in an entirely different location demonstrates that information exists distinctly from the material.
What all these transcendentals imply is that reality can at least be divided into two distinct categories: matter/energy and cognitive realities. Classicly, such cognitive realities are called spirits or souls.
This demonstrates that materialism, and therefore atheism, cannot be true, but it also leaves some serious questions: how can souls exist? In what way do they exist? How do souls interact with matter/energy (since clearly they do)? Can a soul exist seperately from matter/energy? Is there only one soul, or many?
Simply saying "God" does not answer these questions, of course, anymore than saying "parents" answers the questions of my anatomy. Origin does not equate with description. However, I would say that the existance of God, an ultimate cognitive reality, is a good parsimonious starting place for approaching these questions.