The argument from contingency attempts to argue that God exists from the contingency of things. In essence, the argument attempts to address the question "Why are things the way they are?", or more generally "Why is there something instead of nothing?" It goes something like this:
- A1 Everything that exists has an explanation of its existence (either in the necessity of its own nature or in an external cause).
- A2 If the universe has an explanation of its existence, that explanation is God.
- A3 The universe exists.
- T1 The universe has an explanation of its existence. (from A1 and A3)
- T2 Therefore, the explanation of the universe’s existence is God. (from A2 and T1)
A1 is extremely doubtful, especially as applied to the collection of what exists.
A2 is simply incorrect, as we can imagine a wide assortment of possible explanations. If "universe" is taken to mean "existence", it isn't clear how there can even be an explanation.
A3 is a generally unproblematic assumption, except when it is understood to mean that existence is subject to the laws that describe part of existence.
It should be noted, however, that science does not currently provide us with good answers to the above questions. Religion, of course, doesn't either. It is doubtful if such answers are even possible; a final explanation of existence seems to be a logical impossibility, as there is nothing outside existence that can explain it.
On Contingency [A1]
The heart of the argument is the denial of true contingency. Everything must have an explanation - that some fact holds means that it holds because of its own nature (necessarily) or because it was brought about by some external cause (it is contingent on that cause). It is the belief that "everything happens for a reason", that there is actually sufficient (and, indeed, good!) reason why this or that has happened.
There is a third way - no cause, which is true contingency. We often think of things as contingent in the sense that they could have been different. This is related to the issue of free will, the feeling that we could have chosen differently; but also to a more general intuition that things could have been different. But this must be taken seriously, all the way - if they could have been different, then there must be no reason why they aren't different. Formally, they are what they are randomly, not due to any cause. This is how our universe behaves at the fundamental level, or at least that's what quantum mechanics tells us. Things are the specific way they are due to no cause, they are truly contingent (albeit under certain constraints, such as on-average conservation of energy and momentum and so on). This is true contingency.
Physics can be seen as describing the nature of things. In this mindframe, all things that exist do so since it was in the nature of their antecedents to spawn them (or change into them, or so on). Necessarily so according to classical mechanics, or only contingently according to quantum mechanics. The fact that different objects affect each other should not be seen as undermining their nature; it is part of their nature. An object moves in a straight line when not subject to any force as this is part of its nature; but it is also in its nature to change its course when force is applied to it. The appearance of the ball at its destination is never due to its nature alone, as if it was isolated from all of existence; it is due to its nature and the nature of all things around him working together, due to all of Nature. Thus, the dichotomy posed in (A1) between existing due to their nature or due to an external cause is a false one - things exist as they do due to both their nature and the nature of other things, due to the nature of Nature.
- "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) 
From this understanding of physics it is clear that even if everything that exists at any moment has an explanation to it (i.e. there is no contingency), it does not follow that everything taken together has an explanation to it. This is the fallacy Russel alludes to (the fallacy of composition).
It is difficult even to think of what can possibly be the explanation for all of existence, for nothing could exist to provide such an explanation. Some think that existence is in some way cyclical, so that it breeds itself; this is the latest view of some cosmologists, and corresponds to the theistic view that God somehow necessitates its own existence (see the Ontological Argument). But that does not seem to be a satisfactory answer to why the universe exists in this cyclical form to begin with - could it not have been otherwise? The many worlds interpretation of quantum mechanics, when coupled to quantum cosmology, suggests that perhaps all possible universes exist, and ours is but a part of existence. In this case the answer would be something like "Yes, and it is otherwise!". But even this wild speculation doesn't answer the question of why that is so, why all possibilities exist and why there are only specific possibilities that can exist (our world - yes; a world where the symmetries of nature are different - no, not according to modern quantum cosmology). There doesn't seem to be a possible answer to the question of why things are ultimately the way they are. As Russel said, "the universe is just there, and that's all"; there is no way to understand why things exist as they do, and all we can do is describe existence. Perhaps this is due to our lack of understanding of what "existence" means, and when we finally sort it out we'll discover that all things that can exist do, and that this is the only possibility. But we are not in a position to make that argument yet.
In conclusion, the idea that every thing that exists has an explanation for its existence is completely unfounded. Even if it was founded, this would not imply that everything (existence) has an explanation for its existence, which seems to be a logical impossibility.
Existence Seeps Through [A2]
We have already explained that it isn't possible for existence to have an explanation for its existence. In this section we'll treat the "universe" as signifying not the Cosmos as a whole, but rather merely a part of a greater existence. Thus, the universe is but a part of a multiverse.
It is immediately clear from this formulation that (A2) is wrong. There are infinitely many possibilities to explain the existence of the universe by naturalistic means, and even more so by supernaturalistic means which aren't God.
It may be worthwhile to dwell, however, on refuting one possible attempt at explanation, which is that of Creation. It is not possible that the existence of our universe is explainable by its creation ex nihilo from a lower-level of existence, without altering said existence - existence is all in one, lowest, level. For if it was possible, then our universe will not really "exist", instead it will only be a virtual world without any foundation in the reality underlying it. For every difference in our world there must correspond a difference in the lower, underlying reality (if there is one). This means that our description of our reality is a true description of the underlying reality, albeit possibly a distorted description of only parts of it. Physically, this is related to the conservation of information that underlies all physical processes: whatever reality is, any difference in it is a real difference, it is information.
References and Notes
- This formulation follows William Lane Craig 
- Newton's first law of motion.
- Bretrand Russel radio debate, 1948 
- See, for example, 
- See, for example, J.B. Hartle's accessible exposition on how several coexistent classical worlds are braided out of the single quantum reality, arXiv:0806.3776v3.
- arXiv:0803.0944v1 timidly raises an argument in favor of the idea that our reality encompasses all mathematical possibilities, by showing that essentially any spacetime will look like ours. This is, of course, still far from showing that all possibilities are manifested.
- Not the other way, though - there may be differences in the lower level that do not materialize in our level.