I've answered this argument a lot on this site, but I think it deserves a summary question. My aim is make this a reference for any subsequent "origin" questions.
The Cosmological Argument or Argument from First Cause is the proper form of the common argument that the universe must have been deliberately created, and you can't get "something from nothing". It predates Christianity, as Plato and Aristotle had their own versions.
That's the first issue with the argument: it only attempts to prove the existence of a Creator. It is therefore a deist argument, so when a theist uses it to prove a specific god with no further logic it's a step too far. Keep an eye out for this.
The basic premise that everything finite requires a cause is the least controversial part, but even this isn't rock solid. Possible exceptions are found in quantum mechanics, where particles move about in a probabilistic fashion. Until observed, a particle may be anywhere in a small area, and in a sense is everywhere in the area. When you observe it, it picks one spot and stays there. This is of course a gross oversimplification, but the point is that there's no known force moving the particles around. There may actually be no cause as such, and the universe may be far more spontaneous than we think.
Even "something from nothing" is plausible according to a related theory that "nothing" is really a quantum foam from which matter may emerge. This is purely theoretical at the moment (it makes mathematical sense, but there's not much physical evidence), but it's worth remembering that science is actually considering ways like this in which matter could just pop out of "nothing". It can't be dismissed entirely.
Causality may also be irrelevant if time wasn't linear at the beginning, if it had a beginning. An effect must follow its cause, but this is meaningless if chronological order hasn't settled down yet.
The universe is widely regarded by lay people (who aren't young-earth creationists) to have begun with the Big Bang. This may seem counterintuitive - how can something be created by an explosion instead of destroyed? - but it was no ordinary explosion. All the matter and energy in the universe was compressed into a singularity, a point so small it had no volume at all. (Absurd as this sounds, it happens today with large amounts of matter in the centres of black holes.) Then it expanded outwards, and it's still expanding to this day. Once the matter was in that singularity, nothing was created or destroyed, only distrubuted.
How did the matter get in there? Was the Big Bang the true beginning, or a continuation of something else? We haven't a clue. A god is one hypothesis. Other universes, with their own separate systems of time and space, are another. The quantum foam is an outside chance. Who knows what else we haven't thought of.
I like the idea of a multiverse, an eternal group or series of universes setting each other off. It's got one up on gods because it's multiple instances of a known object. We know there's at least one universe (this one), while we don't have a single example of an established god. If you see a huge cabbage patch where the whole crop's been eaten, and you find one fat little rabbit in the corner, do you assume that Bigfoot must have done most of the damage? No, you wonder where all the other rabbits are hiding.
The theory of expansion and contraction, of many Big Bangs and Big Crunches, has fallen apart recently with the discovery that the expansion of the universe is apparently accelerating. That means it will never return to the singularity, and it is not cyclical in the way we thought. That doesn't stop it from being cyclical in other ways, for example stretching until it tears a hole and then draining out to somewhere else.
The point is that if you do accept that everything finite must have a cause, something must be eternal. Either it's the universe/multiverse, or it's a god. There are many theories, and potentially many more, which allow for an eternal universe which needs no cause. Therefore an eternal god is not the only option, and anything which says so is a poor attempted proof of its existence.
Note: The argument that a god created the universe based on the universe's nature, order, awesomeness, etc. is not related to causality. It's the Argument from Design, which is next on the GBA hitlist.