Spacetime can dynamically evolve in a way which apparently violates special relativity. A good example is how galaxies move out with a velocity v = Hd, the Hubble rule, where v = c = Hr_h at the de Sitter horizon (approximately) and the red shift is z = 1. For z > 1 galaxies are frame dragged outwards at a speed greater than light. Similarly an observer entering a black hole passes through the horizon and proceeds inwards at v > c by the frame dragging by radial Killing vectors.
The Alcubierre warp drive is a little spacetime gadget which compresses distances between points of space in a region ahead of the direction of motion and correspondingly expands the distance between points in a leeward region. If distances between points in a forwards region are compressed by a factor of 10 this serves as a “warp factor” which as I remember is $w~=~1~+~ln(c)$, so a compression of 10 is a warp factor 3.3. The effect of this compression is to reduce the effective distance traveled on a frame which is commoved with the so called warp bubble. This compression of space is given by $g_{tt}$ $=~1~-~vf(r)$.
Of course as it turns out this requires exotic matter with $T^{00}~<~0$, which makes it problematic. Universe is also a sort of warp drive, but this is not due to a violation of the weak energy condition $T^{00}~\ge~0$. Inflationary pressure is due to positive energy. The gravity field is due to the quantum vacuum, and this defines an effective stress-energy tensor $T^{ab}$ with components $T^{00}~=~const*\rho$, for $\rho$ energy density, and $T^{ij}~=~const*pu^iu^j$, for $i$ and $j$ running over spatial coordinates $u^i$ velocity and $p$ pressure density. For the de Sitter spacetime the energy density and pressure satisfies a state $p~=~w*\rho$ where $w~=~-1$. So the pressure in effect is what is stretching out space and frame dragging galaxies with it. There is no need for a negative energy density or exotic matter.
Negative energy density or negative mass fields have serious pathologies. Principally since they are due to quantum mechanics the negative eigen-energy states have no lower bound. This then means the vacuum for these fields is unstable and would descend to ever lower energy levels and produce a vast amount of quanta or radiation. I don’t believe this happens. The Alcubierre warp drive then has a serious departure between local laws of physics and global ones, which is not apparent in the universe or de Sitter spacetime. The Alcubierre warp drive is then important as a gadget, along with wormholes as related things, to understand how nature prevents closed timelike curves and related processes.
Addendum:
The question was asked about the redshift factor and the cosmological horizon. This requires a bit more than a comment post. On a stationary coordinate region of the de Sitter spacetime $g_{tt}~=~1~-~\Lambda r^2/3$. This metric term is zero for $r~=~\sqrt{3/\Lambda}$, which is the distance to the cosmological horizon.
The red shift factor can be considered as the expansion of a local volume of space, where photons that enter and leave this “box” can be thought of as a standing wave of photons. The expansion factor is then given by the scale factor for the expansion of the box
$$
z~=~\frac{a(t_0)}{a(t)}~-~1
$$
The dynamics for the scale factor is given by the FLRW metric
$$
\Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi G\rho}{3}
$$
for $k~=~0$. The left hand side is the Hubble factor, which is constant in space but not time. Writing the $\Lambda g_{ab}~=~8\pi GT_{ab}$ as a vacuum energy and $\rho~=~T_{00}$ we get
$$
\Big(\frac{\dot a}{a}\Big)^2~=~H^2~=~\frac{\Lambda}{3}
$$
the evolution of the scale factor with time is then
$$
a(t)~=~\sqrt{3/\Lambda}e^{\sqrt{\Lambda /3}t}.
$$
Hence the ratio is $a(t)/a(t_0)~=~ e^{\sqrt{\Lambda /3}(t-t_0)}$.The expansion is this exponential function, which is Taylor expanded to give to first order the ratio above
$$
a(t)/a(t_0)~\simeq~1~+~H(t_0)(t_0-t)~=~1~+~H(t_0)(d-d_0)/c
$$
which gives the Hubble rule. $z~=~a(t)/a(t_0)~-~1$. It is clear that from the general expression that $a(t)$ can grow to an arbitrarily large value, and so can $z$. On the cosmological horizon for $d~-~d_0~=~r_h~=~\sqrt{3/\Lambda}$ we have $z~=~1$.
Looking beyond the cosmological horizon $r_h~\simeq~10^{10}$ly is similar to an observer in a black hole looking outside to the exterior world outside the black hole horizon. People get confused into thinking the cosmological horizon is a black membrane similar to that on a black hole. Anything which we do observe beyond the horizon we can never send a signal to, just as a person in a black hole can see the exterior world and can never send a message out.
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
In any curved spacetime we can still talk about local reference frames that are small enough scale we can ignore the curvature. We also can ask if there are closed timelike curves (CTC) which basically is asking whether we can time-travel to our past selves. CTCs are strongly thought to be impossible in reality.
The universe is thought to be spatially flat, but the spacetime as a whole is curved. CTC's are impossible: at each point in spacetime you have an "age of the universe". To be precise, this is maximum path-length (proper time) a geodesic could have between the big-bang singularity and said point. Any time-like or light-like curve is moving in the direction of increasing age of the universe; this is just as strong a concept of "future" and "past" as in flat spacetime.
With a single warp-drive you don't have CTC's. But you can still get CTC's with multiple warp-drives. Suppose you build a warp-drive on Earth and send it out into space. You start with an (almost) flat initial-condition and then generate a strongly curved spacetime (your warp bubble). Starting from a flat spacetime (or for very large scales from the spacetime of the universe), is much more physically realistic than starting from any other spacetime. You have to make your weird and wonderful curvature from an "empty canvas" !
With a warp-bubble, the highly curved spacetime is on a small scale. This allows us to glue two bubble spacetimes together so long as the ships don't get very close to each-other. If we consider two Earths, moving relative to each-other, that each make a warp-drive, we can set up the system to generate CTCs. This is one reason we suspect this to be impossible.
There is another reason to suspect making warp drives is impossible: Geodesics would have to diverge in some region, which is an anti-gravity effect. Neither matter nor light can make anti-gravity (antimatter has positive mass just like matter). The "attractive gravity only" rule is more precisely defined as an energy condition and at least one of these is violated by warp drives. Violating certain energy conditions would make the speed of sound faster than light which also allows for time-travel paradoxes.
In general, no known solution with CTC's is physically realistic. They either involve infinitely large systems that cannot be setup from an "empty canvas" or violations of energy conditions. For example, the Kerr metric concentrates it's energy condition violation in it's singularity. Real black holes are thought to lack this feature and be much deadlier instead.