First of all, definitions. Cosmology is the study of the origins of the cosmos, that is the whole universe. The cosmological argument for the existence of God then is simply that the origin of the cosmos requires explanation, and that God is the best explanation for it. However that is merely what the argument attempts to show, not the argument itself. We shall see that the argument can actually be presented in several different forms, and we will deal with each in turn.
UNDERSTANDING THE ARGUMENT
Argument from Contingency: Thomas Aquinas
While there were some precursors to Aquinas' presentation of this argument, Aquinas was really the first to pen it as such. Aquinas presented the cosmological argument in 3 different forms. The first two, the argument from movement and the argument from causation, are really the same thing, the latter merely being an abstract form of the former. We will look at this in the next section.
The third form he presented was the argument from contingency. The flow is as follows:
P1: All contingent beings require a source for their existence
P2: If everything were contingent, than nothing would exist
C1: Thus there exists at least something which is necessary, from which ultimately comes things which are not.
C2: God is that necessary being.
The ultimate problem with the argument as Aquinas puts it is that it is incomplete. The move from C1 to C2 is forced. Additionally, it doesn't deal with the possibility that the universe itself is contingent. For the first, he primarily refers to the argument from causation. However, as to the second, I believe that to some degree Aquinas here was primarily interested in the existence of humans and life, as opposed to rocks, planets and stars.
We can extract this out and think about this in a more modern way. There exists material in the world, in the sense of atoms and energy. There are certain things which are simply the formation of these particles due to the natural interaction of forces. However, the fact that some of this material gathers in such a manner as to become something greater than its constituent parts, such as a human, requires explanation. Planets and stars to necessary manifestations of physics, but the development of human is physically odd. It is demonstrable that the development of life is something which is incredible difficult, even if we can develop from natural causation. Therefore, unlike rocks, planets, and stars, we require special explanation.
This is why he distinguishes from necessary beings which come from other necessary beings, like rocks and planets, versus necessary beings which have no cause, like God. Ultimately God is the ultimate cause for all contingent beings because he is a being without cause, however, that goes to the argument from causation which we will discuss below.
But before we move on to that, it is important to note how much more potent this argument becomes when understood in light of the big bang and eventually universal death. These two events act as a kind of merism for the whole of the universe, showing that the universe itself (i.e. the rocks, planets and stars) is contingent and thus requires explanation.
God is thus offered as a parsimonious answer to such a question. I shall discuss the parsimony of God as an answer in the final section of this post.
Argument from Causation: A Simple Matter of Time
The argument from causation is what most people refer to when they refer to the cosmological argument. This is mostly because it is considered the superior form of the argument. However, the argument has changed over the course of the years.
Aquinas presented the argument as a series of events: every thing moves, but each movement is in fact caused by something else. This is further extended to the nature of causation, where every cause has an effect. He then makes a very important move: he insists that the concept of infinite regress is illogical. Therefore, for every chain of causation, there must be a beginning place which was not caused or moved, but simply is. Due to the interconnectivity of causation, recognizing events seems to have a multiple interrelated causes, it is considered parsimonious to say that all lines of causation have a single ultimate cause. Aquinas simply names this ultimate cause God (again a conclusion which is a bit forced).
In parallel, this argument was also presented in Islamic circles, specifically of the Kalam tradition, but formulated slightly differently. Instead of focusing on a line of causation, they instead consider the ontology of a thing, in that anything which has a beginning must have a cause. While Aquinias's arguments do not assume the beginning of the universe but instead focus on the natural operations of it, the Kalam argument does assume the beginning of the universe, and states that because the universe began, it must have a beginning. The syllogism is pretty airtight:
P1: Everything which has a beginning has a cause
P2: The universe has a beginning.
C1: The universe has a cause
Again in a forced move, God is offered as this cause.
I personally would expand on this, focusing on Aquinas's presentation. If we consider a moment to be an event, and that moment to have been brought into existence by the last moment (considering the recent theory of the discrete nature of time, this a rather accurate way to think of it), we can then see the full timeline to be a chain of causation. Therefore, we agree with the premise that infinite regression is categorically false, we can then conclude that time itself must have a beginning, and that it is an aspect of the universe. Therefore we can expand the Kalam definition and say that anything which is bound by, or exists within time must have a cause, but it existence is an aspect of this causation chain.
Therefore, when we consider what this ultimate cause is, we conclude that is must be something which is not bound by time. The Christian view of an eternal God anticipates such a quality, instead of being formed by it, further establishing God as a solution.
CRITICIZING THE ARGUMENT
There is a very legitimate criticism to his argument,and it has already been mentioned several times, but I want to treat it last. First I want to deal with three criticisms which I feel do not really work: rejection that contingents need causes, acceptance of infinite regress, and the proposition of a cause for God.
First let us deal with the principle premise of the argument: that things with a beginning require a cause. Many Atheists have argued that within quantum mechanics, things often pop into existence without cause. Therefore the tautology that all things with a beginning require a cause is defeated because it assumes Newtonian physics.
There are two responses to this. First of all, quantum mechanics has more unknowns than it has knowns. It is an incredibly controversial area of science. What may seem like a particle coming into existence without cause may simply be a particle coming into existence by an unknown cause. It is premature to declare victory from such an example. But it is equally premature to say that therefore the point is completely illegitimate. This brings me to my second point: the rules of quantum mechanics do not operate within the macrocosm. Big Bangs are not occurring within the universe all the time. No one is concerned about the sudden appearance of a singularity within the solar system. This kind of event isn't happening. However, that is precisely what such an argument is presupposing. The commencement of the macrocosm is an event within the macrocosm, and as such a quantum level event would seem to have no baring on it.
Now for the second criticism: the acceptance of infinite regress. One of the major moves that the cosmological argument makes is that an infinite chain of causations is an absurdity. However, one could say, "Why? Let's just believe in infinite regress." Well, first of all, one could say anything. I could say, "Why can't we have a square circle?" At some point, we need to assume some things. This is a basic fact of logic: we need to start somewhere. Therefore, we need to consider the fact that the notion of infinite regress certainly seems absurd. Even those that advocate otherwise recognize that the notion is at least ineffable. Therefore, I would argue that those who propose infinite regress need to first show that it is a possibility, and they have yet to do so.
The final criticism is that the proposition that God himself also requires a cause. However, simply from my presentation one can see why such a criticism is ridiculous. Whether we accept the Kalam premise (that all things with a beginning have a cause) or my own (all things bound by time have a cause), one can see that neither premise would apply to the articulation of God proposed by Christians. We hold that God is eternal (which is different than infinite BTW) and does not exist within time. As such, this criticism is nothing more than a straw man.
It is admitted that Aquinas does not offer explanation as to why God is exempt from his arguments. I would assume though that he would reference both God's eternalness, and His immutability.
There is a real flaw to the cosmological argument: there is no deductive reason why God has to be the answer to causation. While God is in fact an answer, and a good answer at that, it may be possible for there to be other answers.
There has been three proposed possibilities for the ultimate cause: A single eternal omnipotent God, several gods, and the multiverse. Here we are including the Jewish, Christian, Muslim, and Deistic views of God as a singular possibility, for they differ in character, rather than ontology. As to a single God, verses several gods, we can say that a single God is more likely due to parsimony (the inductive principle that the simplest answer is the more likely answer). This leaves us with God vs. the multiverse.
Here it is important to point out that both are equally unfalsifiable, so neither can assessed scientifically, only philosophically. Any favor granted to the multiverse for being more "scientific" is based solely off of it being an atheist position, which is either begging the question or special pleading (or both).
The hypothesis of the multiverse (I refuse to consider this to be theory) is based to some degree on a thought experiment on the nature of dimensions, but being able to fathom greater dimentions does not demonstrate those dimensions to not be flat. The concievability of the multiverse does not demonstrate the existence of the multiverse. To another degree it is also part of string theory which itself is quite controversial.
It is also important to note that nature of the multiverse is barely proposed. It appears to be a place where realities pop into existance, and either manage to sustain themselves or fail. However, how are these realities able to exist side by side? What mechanism within the multiverse causes their creation?
To what extent can we say the law of entropy to be at work here, since that is the problem with most singular reality cosmological models.
Additionally, the multiverse is a realm of caused contingent realities. Nothing within the multiverse seems to be necessary, or seems to be able to give account for the commencement of a new reality. A collection of contingents requires explanation as much as a single contingent, and the multiverse as it is currently articulated, is defined by its contingent members, rather than by the environs within which this members exist. Therefore it doesn't seem to directly address any of the points of the cosmological argument.
This is compared to the articulation of the Christian God: incorporeal, ontologically simple, eternal, and necessary. Additionally, God is a cognitive reality, and as such can account for the commencement of something new. The nature of thought can account for the transition of the non-existence of reality to the existence of it better than some vague mechanism of a multiverse. As such, God seems to be a superior answer to the cosmological argument than the multiverse, if we can consider the multiverse even satisfactory.
For more on the cosmological argument, I highly recommend the site of William Lane Craig, who re-popularized the argument back in the 70s: http://www.reasonablefaith.org/popular-articles/existence-and-nature-of-god