What is the quantum computational model? I have occasionally heard people talk about quantum algorithms and about states and the ability to consider multiple possibilities at once, but I have never managed to get someone to explain the computational model behind this. To be clear, I am not asking about how quantum computers are physically constructed, but rather how to view them from a computational point of view.
I'll echo Martin Schwartz's recommendation of Nielsen &amp; Chaung as the standard reference; there are many others as well. 

Research in the field prefers to consider uniform families of quantum circuits, which (ironically) are directed acyclic networks describing how the state of one or more registers transforms with time, in a way similar to classical boolean circuits. If you wish to learn more, I recommend learning in terms of this model.

I would like to give some qualitative answers to complement Martin's response.


Quantum computation does not actually consider "multiple possibilities at once" --- or more precisely, whether or not you consider them to consider multiple possibilities at once is a matter of your choice of interpretation of quantum mechanics, i.e. a philosophical choice which has no bearing on the ability or predictions of the computational model. ("Considering multiple possibilities at once" corresponds to the "many worlds interpretation" of QM.) 
At the very least, one can say that a quantum computer considers multiple possibilities at the same time only to the extent that a randomized computation using coin-flips considers multiple possibilities at the same time. This is because:
Quantum states are generalizations of "the usual" probability distributions --- with some simple but important differences. A probability distribution can be represented as a non-negative real vector whose entries sum to 1: that is, a unit vector in the &#8467;1 norm. Probabilistic computations must map &#8467;1-unit vectors to other such vectors, and so they are described by stochastic maps. One can describe quantum computation in a similar way, except using &#8467;2-unit vectors over &#8450; (not restricted to be real or non-negative); transformations are by those maps which preserve the &#8467;2-norm, i.e. unitary operations. This difference is not trivial, of course, nor does it explain yet what the coefficients of the quantum state vectors mean. But it may help to explain what is going on with Hilbert spaces and tensor products in quantum computation: to wit, exactly the same things as happen in probabilistic computation. The configuration space of a random bit is a vector in &#8477;+2 (where &#8477;+ are the non-negative reals); but because random bits can be correlated, we combine the configuration spaces of one or more random bits by taking the tensor product. So the configuration space of two random bits is &#8477;+2&nbsp;&otimes;&nbsp;&#8477;+2&nbsp;&cong;&nbsp;&#8477;+4&nbsp;, or the fully-general space of probability distributions over the four distinct two-bit strings. An operation A on the first of these random bits which does not act on the second is represented by the operator A&nbsp;&otimes;&nbsp;I2&nbsp;. And so on. The same constructions apply to quantum bits; and we can consider quantum registers over sets of distinguishable elements the same way we consider probability distributions over such sets, again using &#8467;2-norm vectors over &#8450;. This description actually describes the "pure" quantum states --- the ones for which you can in principle transform in an information-preserving way to a delta-distribution over the bit-string 00...0 (or more precisely, to a state arbitrarily close to this in the &#8467;2 norm). On top of the quantum-randomness (of which I have not yet mentioned anything explicit), you can consider vanilla-convexity-randomness corresponding to probabilistic mixtures of quantum states: these are represented by density operators, which can be represented by positive definite matrices with trace 1 (again generalizing "classical" probability distributions, which may be represented by the special case of positive diagonal matrices with trace 1). What is important about this is that, while quantum states are often described as being "exponentially large", this is because they are usually described using the same mathematical structures as probability distributions; why probability distributions are not described as "exponentially large" in the same way is unclear (but ultimately unimportant). The difficulty of simulating quantum states come from this fact, together with the fact that the complex coefficients of these &#8467;2-distributions (or the complex off-diagonal terms of the density operators, if you prefer) may cancel in a way that probabilities cannot, rendering estimation of them more difficult.
Entanglement is just another form of correlation. For probabilistic computation on e.g. boolean strings, the only "pure" states (which can be mapped by information-preserving transformations to a delta-peaked distribution on 000...0) are the "standard basis" of delta-peaked distributions on the different boolean strings. Thus, this basis of &#8477;+2n is distinguished. But there is no such distinguished basis in quantum mechanics, as far as we can tell --- this is clearest for quantum bits (look up spin 1/2 particles, if you want to know why). As a consequence, there are more information-preserving transformations than just the permutations: a continuous group of them, in fact. This allows would-be quantum computers to transform states in ways which are not possible for probabilistic computers, possibly obtaining an asymptotic advantage over them. But what about entanglement, which many people find mysterious, and claim to be the cause of the speed-up of quantum computers over classical? "Entanglement" here is really just a form of correlation: just as two random variables are correlated if their distribution is a convex combination of more than one product distribution (with different marginals on each variable), two "quantum variables" are entangled if their distribution is a linear combination (with unit &#8467;2-norm) of two valid product distributions; it is the same concept under a different norm, and plays a similar role in communication tasks. (For example: "quantum teleportation" in quantum communication corresponds to encoding and decoding a message using a one-time pad classically.) This is a form of correlation which is more general than just classically correlated bits; but the only way to show this is that the correlations encoded in the entangled state apply to more than just one privileged basis. In a manner of speaking, entanglement is a consequence of the absence of a privileged basis.  People like to invoke entanglement as the key element of quantum computation, but this simply doesn't seem to hold water: there have been results showing that entanglement is not quantitatively important for Shor's algorithm to factor large integers, and that indeed a quantum system can have too much entanglement to be useful for a computation. In fact, everywhere that I am aware of that entanglement plays an important role in a quantum protocol is essentially one of communication (where correlations would be expected to play an important role for a classical protocol).


At this point, I begin wading into the domain of personal opinion, so I'll stop here. But hopefully, these remarks might de-mystify some of what is obscure about quantum computation and how it is described.