Causality means that for any two events A,B, there has to exist an ordering that says whether A can influence B or B can influence A, and in the normal examples with "time", the ordering is the condition
$$ t(A) < t(B). $$
If the condition above is satisfied (i.e. if A precedes B), then A may influence B.
An ordering - a transitive relation - has to exist in order to avoid logical contradictions. If the relationship were not transitive, for example, it would be possible to find triplets of events A,B,C such that A influences B, B influences C, C influences A. That would be a "closed time-like curve" and it would lead to logical inconsistencies because in general, there would be no way to choose the outcomes of the events A,B,C so that all the three implications are preserved.
Those contradictions are avoided in any causal theory because the outcome of event A (in a causal and deterministic theory, to be specific) is never calculated from conditions at event B if A is the cause of B (if A precedes B). It's the other way around. Causality makes it clear which data are "inputs" and which data are their "outputs", so because of this orientation, there can't be any contradiction.
In a geometric setup, the comparison of a coordinate associated with the events is the only way how to produce ordering as a relationship. We call this coordinate "time".
In special (and similarly general) relativity, the condition for A to be able to influence B becomes sharper - $t(A)<t(B)$ has to hold in all reference frames which means that B has to belong to the future light cone of A.
Edit regarding 3+1 spacetimes and causality
I'll keep adding to the answer as I get more information, and hopefully everything will just evolve along. At the very least, I'll have a set of notes to work from in the future :) This is also the first, broadest, cut at an actual answer regarding causality.
Alcubierre sets out to find his warp drive metric using a 3+1 formulation of spacetime. In the 3+1 formulation, spacetime is described as a set of constant coordinate time spacelike hypersurfaces, (foliations, for the fancy). In doing this, you wind up with a line element that looks like (see erudite comments from @Jerry Schirmer below, I'm playing catchup):
$ds^2 = -d\tau^2 = \gamma_{ij}dx^idx^j + 2\beta_i dx^i dt - \left(\alpha^2 - \beta_i\beta^i\right)dt^2$,
where $\alpha$ is the lapse function, and is positive, and $\beta$ is the shift vector between spatial foliations. $\alpha$ describes how quickly time evolves, while $\beta$ describes how the spatial coordinates evolve in time. In other words $\alpha$ and $\beta$ describe how your ship moves through space and time per incremental step.
What's important here is that $ds^2$ is positive and for real space, $\gamma_{ij}$ is as well. Remember, hyperbolas look like $\dfrac{x^2}{a^2} - \dfrac{t^2}{b^2} = 1$. So, the line element equation above describes a globally hyperbolic system in space time. What's that mean? It means you can't close a curve in spacetime, so you can't violate causality. Note that $\beta^i$ squares up where it's important to maintain sign to maintain a hyperbola. I'd think there should be another requirement that $\alpha^2 > \beta_i\beta^i$, but Alcubierre doesn't mention this, so I'm guessing we don't actually need it.
Alcubierre isn't done yet, he's still got to find a metric that will fit in a 3+1 spacetime and do what he wants, (provide faster than light propulsion), but if he does, the above property of 3+1 spacetimes will guarantee causality.
Edit
I Stand Corrected Regarding the Alcubierre Drive
@Superbest pointed out, that the claims for the drive were that it could go faster than the speed of light with regard to the laboratory frame, and hence with laboratory velocity. I found the original paper by Alcubierre on arxiv[2], and...
he's absolutely right!
The paper is amazingly well written and folks that have had a grad level general relativity class should be able to easily traipse through it. Alcubierre even shows that causality won't be violated. I haven't had time to digest the material enough to say why causality isn't violated except with the very unsatisfying statement, "Well, the math works out." Alcubierre was also quick to point out that he felt that with a bit of effort he could come up with an example that would violate causality:
"As a final comment, I will just mention the fact that even though the spacetime described by the metric (8) is globally hyperbolic, and hence contains no closed causal curves, it is probably not very difficult to construct a spacetime that does contain such
curves using a similar idea to the one presented here."
OK, so to summarize. The math explanation and associated formulas I wrote below are correct. With uniform acceleration and no exotic matter whatsoever, you can travel more than x light years in x proper time years. In the case of the Alcubierre drive, however, that's not the trick they're playing. I hope to have more details soon, but in the meantime I'll leave you with this quote from Schild regarding the twin paradox and general relativity.
"A good many physicists believe that this paradox can only be resolved by the general theory of relativity. They find great comfort in this, because they don't know any general relativity and feel that they don't have to worry about the problem until they decide to learn general relativity."
End Edit
The explanation given in the Washington post article triggers a pretty common misconception:
"If an object reaches a distance x light years away in under x years, then it must be travelling faster than the speed of light."
What the article failed to mention is that the 14 days quoted is in the reference frame of the ship. The equation for the distance travelled with respect to time in the frame of the ship, (known as proper time), is
$$\mathrm{distance} = \dfrac{c^2}{a}\cosh\left(\dfrac{at}{c}\right)-\dfrac{c^2}{a},$$
where $a$ is the acceleration of the ship and $c$ is the speed of light.
Using this formula, it can be shown that at an acceleration of 188g, (188 times the acceleration due to gravity), the ship could reach Alpha Centauri in 14 days of ship time. You might point out that 188 g's would surely smush everyone against the back wall of the ship, but the beauty of the theoretical drive described is that you carry your own gravity well along with you and therefore, you're always in freefall and don't feel the acceleration.
Here's the problem though. The time that will have elapsed here on Earth will be much, much greater than the 14 days that elapsed on the ship. The expression for the time elapsed on Earth is
$$\mathrm{Earth\ time\ elapsed}= \dfrac{c}{a}\cosh\left(\dfrac{at}{c}\right),$$
which can be used to show that when the ship reaches Alpha Centauri, 817 years will have passed here on Earth.
The calculations shown here are nothing new, by the way. Rindler applied them to the problem of relativistic space travel for the first time in 1960 in a Physical Review article titled "Hyperbolic Motion in Curved Space Time" [1].
References
Rindler, W., "Hyperbolic Motion in Curved Space Time", Phys. Rev. 119 2082-2089 (1960).
Alcubierre's original warp drive paper
http://arxiv.org/abs/gr-qc/0009013v1
Best Answer
Suppose you and I have a conversation from a long distance away. We're at rest with respect to each other and communicate much faster than light. I say "How are you", and you wait a short time and say, "I'm fine thanks."
From our point of view, you were responding to my question. However, from a reference frame moving from me to you at relativistic speed, your clock is significantly ahead of mine (a relativistic effect). This means that although you thought you received the message shortly after I sent it, in this frame you didn't. You actually received the message at an earlier time (before I sent it), but you thought it was later because your clock is ahead.
From your and my point of view, the order of events is
From the frame moving from me to you, the order of events is
The fact that the order of events changes between reference frames is simply part of relativity, with or without faster-than-light communication. However, it seems strange in this scenario because you are responding to me. Presumably, if I had said, "Where are my car keys?", you would have chosen a different response than "I'm fine thanks." How then is it possible that you responded to my greeting before I uttered it, at least in some frame?
I'm not sure if this "violates causality", but it's unintuitive.