Cosmological arguments attempt to show that a god is necessary from principles of causation. There are several such arguments (and other related ones), including the Argument from First Cause, the Kalam Cosmological Argument (advanced by William Lane Craig), and the Argument for an Unmoved Mover. The argument generally follows this rationale:
- A1 Everything is caused by something else, creating a - possibly infinite - chain of causation
- A2 An infinite chain is impossible, and the only alternative is to cut it by a First Cause
- A3 This 'First Cause' is God
All three steps are problematic. It is eminently not clear that everything is caused by something, and although there are variations that might be more plausible it isn't clear how they relate to existence as a whole. It is not clear why an infinite regression could not be the ultimate description of reality, or that a First Cause is necessary for a finite chain, and so the supposition of a First Cause is suspect at best. Even if both prior assumptions are accepted, it is not at all clear why this 'First Cause' should be called 'God'. Rejecting each claim in turn is the task of the following, more detailed, analysis.
For a simpler summary of the argument see Uncaused cause.
On Causal Chains and Other Infinite Regress [A1]
The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. - Thomas Aquinas, Summa Theologica, First Part, Question 2, Article 3 (13th century)
Notice that Aquinas is talking about change in general, saying that any change requires the object to have the potential to undergo that change and to be driven to that change by the effect of something already in that state. This is not what our modern physics tells us. A physical system must have the potential to be in some state in order to acquire it (technically, the state must exist in its phase space), but movement to that state can be driven by other means. A simple way of seeing it is by considering the conservation of energy in its many forms. A movement can be started due to potential gravitational energy (height), electric energy (an object splitting into two parts due to electric repulsion), and so on. Indeed, quantum mechanics even implies that there is no need for sufficient energy for sufficiently brief times; just the potentiality suffices.
But notice that this doesn't affect the argument. It doesn't matter for the rest of the argument that it is specifically movement (or, generally, change) that results in regress; what matters is that something results in regress. For this reason many similar arguments have used other regressions. Aquinas himself uses the regress of efficient cause (refuted by quantum mechanics) in his second proof, and the Kalam Cosmological Argument uses a similar "everything that begins to exist has a cause". Perhaps one formulation or the other is incorrect; but is there is at least one formulation that is correct?
It does appear so. Our physical theories do say that every physical state is precded by a previous one, from which it evolves according to the laws of nature. The issue gets complicated, since General Relativity suggests that "time" is essentially a dimension, so the question is transformed into one of boundary conditions. This reveals that there are actually more options than two - our universe "evolves" in space as much as it does in time, and perhaps those segments of existence who are shaped right have "time" so time begins but nevertheless the chain goes on ubroken; it is possible that time is circular, so that the chain repeats itself; it is possible that time is truly infinite; it is possible that time begins but does not end, or ends but does not begin, or both begins and ends; it is even possible that there are no well-defined boundary conditions. All of this (and doubtless, more) is possible, although perhaps not all options are equally plausible at this point. Despite this subtlety, the core idea that there is a regression that the laws of physics commit to and is hence very plausible does stand firm.
So it seems is a very good case to be made for regress. This does not mean that the proposition is true - quantum mechanics showed us that efficient causes don't follow each other in a chain, and that was a big shock, so perhaps we're in for another and we'll discover regression isn't real. And it must be realized that this regress is within what exists, not about it. As Bretrand Russel said,
- "I can illustrate what seems to me your fallacy. Every man who exists has a mother, and it seems to me your argument is that therefore the human race must have a mother, but obviously the human race hasn't a mother -- that's a different logical sphere." - Bertrand Russel (1948) 
But to the best of our current knowledge, some version of regress does appear to be a reasonable belief.
Cutting Infinite Chains [A2]
The cosmological argument rejects the possibility of an infinite chain of regress, and posits that the only possible alternative is an Unmoved Mover, First Cause, or so on - some element of reality that is not answerable to the first assumption (A1) yet moves (or causes, or so on) the first link in the chain that can then proceed according to the normal rule (A1).
The above discussion should make it clear that there are other alternatives, such as cyclical time. It is also not necessary for the chain to have a clear first element of this sort - consider pair production, whereby two particles are formed out of energy and quantum fluctuations. Each then proceeds to serve as efficient cause for a host of other events, but there is no event or particle or much of anything that can be said to be the First Cause of those chains. The chain just appears, starting at some point but with no real defined first element in it.
This is tied to another issue, which is that our physical theories acutally strongly suggest that there are infinite actual chains. Part of the problem of the above scenario is that you can always get arbitrarily close to the starting point of the chain, there are always physical states and causes preceding any moment and no single event or moment exists just after the moment the chain begins (just like no Real number immediately follows the number 0.0). Indeed, there is an infinite chain of events within any finite amount of time, with any event not directly causing any other event (again, much like 0.0 is not followed by any Real number). The entire picture of a chain is mistaken, since the causal "chain" is infinitely dense.
So it appears that infinite chains are not only plausible, but that even densely infinite chains are plausible. For such chains it may be that there is a first element (mathematically, it is a closed section) or that the first point is outside the chain (an open section); in the latter case, perhaps the theist would like to say that the first element is "transcendent" to the chain. But it is clear that either way having an infinity of elements is unrelated to the question of whether the length of the chain is infinite (whether it goes infinitely long into the past, the boundary conditions) - and that there is no a priori reason to believe that the chain wouldn't go infinitely long into the past (or sides, or so on).
Are there other reasons to think that it won't, and that it won't in such a way that a first element of some sort will exist? It is unclear. If one takes the Big Bang theory within General Relativity seriously, then there is a (possibly transcendent) point at the beginning of time. Hardly anyone does, however, as General Relativity should be supplanted by Quantum Gravity at these conditions and it isn't at all clear what this means. No one, at this time, knows the general geometry of existence. An interesting conjecture is the "No Boundary Condition" , which essentially says that there is no real first element, as the geometry is very much like a sphere's surface (you can go round-and-round a ball, without ever meeting a boundary; it's kinda like the idea of circular time). If it is correct, then the chain is finite but has no first element (much like the circumference of a sphere). Although there is some evidence for this, it is still only a conjecture at this point.
So at this time there is no real reason to accept or to reject the hypothesis that there is a First Point, some element of reality or a point transcendent to reality that in some sense is the beginning of a regress but is not the result of one. It can go either way, but in any case is not related to the arguments against the existence of infinite chains, which hold no water.
Is It God? [A3]
The real problem with the cosmological argument is the presumption that God is in any way related to the arrest of regression. Clearly, the point at the beginning of the pair production, if there is one, is not god. Why would the singularity at the Big Bang be god? Physics describes reality. This description may say that existence begins at some boundary, or perhaps at many boundaries (multi-dimensional geometry is a mess) - so what? This does not, by any means, imply that there is something beyond the described existence. To say that there is you need to make a metaphysical assumption that boundaries on existence are impossible; this seems unfounded.
It is entirely unreasonable to say that something outside existence can even possibly "cause" existence (or "movement", or so on). So even if the "No Boundary On Existence" metaphysical admonition is adhered to, it should apply to all things, god included - so the argument cuts its own leg, simply arguing in favor of a No Boundary condition on existence instead of one with a boundary.
Ultimately, there is no need for the god hypothesis. The transcendent points described by our modern physics do not exist (following the naturalistic interpretation), so need no explanation. For closed chains, the starting point is part of the physical description. Either way, there is no mystery - existence is described to consist of those elements and these behave in such a manner, and that's that. No further description of these points is necessary, and there is nothing in the description that necessitates existence beyond the described one.
It is unreasonable to associate such points with the traditional concept of God. The singularity of the Big Bang is not a Person, it is a spacetime point of infinite density. It is not Omniscient, or Omnipotent, or Perfectly Moral - it is simply a point in the fabric of spacetime.
A1 is doubtful; probably correct for things in the universe in some form, perhaps correct for a universe, and it isn't clear how it can even possibly apply to existence as a whole
A2 makes an incorrect argument, but the existence of a First Cause is possible
A3 is simply not reasonable
The cosmological argument at its heart attempts to answer the question "Why is there something instead of nothing?", or "Why Are Things The Way They Are?". This is a mistaken route to take towards answering these questions, however, as it gets bogged down in questions of causality and inifnities. For a better treatment of these issues, see the Argument from Contingency.
- Think of existence as a golf ball, with "time" existing only within the indentations, each being a separate universe. Each dimple begins, but nevertheless the surface of the ball is continuous. In much the same way, time begins yet the surface of space-time may be continuous.
- Bertrand's Russel discussion 
- For a layman's level lecture, see