Summary: A spaceship using an Alcubierre warp drive would involve assembling a ring of negative energy around the spaceship which would distort spacetime in this particular way: the spaceship sits in a "bubble" of flat, Minkowski spacetime which is "pushed" by expanding space behind it and "pulled" by contracting space in front of it. The spaceship does not move through space, but rather space itself moves and carries along the spaceship for the ride. Since general relativity places no limit on how fast space can move, the space can "carry" the warp bubble and spaceship away at faster than light (FTL) speeds.
This video was produced by PBS Space Time.
Space can push things at FTL speeds
Einstein's special theory of relativity places a "cosmic speed limit" on how fast objects can move through space: thou shalt not move through space faster than the speed of light, \(c=366,000 mps\) (miles per second). In other words, suppose that Alice is in the middle of empty space (far away from any massive galaxies) floating in an elevator; no matter what kind of propulsion system was installed in her elevator, she and her elevator could never move through space faster than the speed of light. But Einstein's general theory of relativity places no speed limit at all on how fast space itself can move. Space can move as fast as it wants, even faster than the speed of light. Since the Alice-elevator system is in the middle of empty space far away from any massive objects, we know that there is no force of gravity moving them through space. And since Alice turned off her elevator's propulsion system, we can be sure that she's not moving through space. With all of that said, what if the space behind her was expanding—faster than the speed of light?
An Alcubierre warp drive—whatever that is—involves a region of flat, Minkowsky spacetime "taking a ride on" an expanding space faster than the speed of light. This is analogous to how Bob in a distant galaxy—tens of billions of lightyears away—is moving away from us earthlings faster than light speed. As the space between everything in our universe expands (which we are pretty sure is due to dark energy), the expanding space "pushes" Bob away from us at faster than light (FTL) speeds. Back to Alice though. Alice is also tens of billions of lightyears away, but her and her elevator are in one of those great cosmic voids where everything is mostly empty space. Since Alice is in an elevator far away from any other massive bodies, then the Alice-elevator system is an isolated system. And since the Alice-elevator system is very unmassive, they would be in a "bubble" of flat, Minkowsky spacetime. Let's say Alice turned off her elevator's propulsion system and that the motion of the Alice-elevator system through space relative to us earthling (represented by their peculiar velocity) just happens to be zero; then, according to special relativity, neither their clocks nor meter sticks will appear to contract or dilate relative to us earthlings. Their situation is this: they ride in a bubble of flat spacetime (with no time dilation or length contraction) along the cosmic waves of expanding space faster than light.
Now, replace Alice and her elevator with a spaceship and a crew, and this starts to sound a lot like an Alcubierre drive (which I promise I'll define at the end): the spaceship and the crew ride in a bubble of flat spacetime (where neither their clocks nor rulers are distorted relative to us earthlings) along an expanding space. But there are two differences between Alice's situation and the crew's: for the crew, the expansion of space isn't due to the Hubble expansion of the universe. This expansion was engineered by the crew. They assembled a ball of negative mass which, according to Einstein's field equations, caused the space "behind them" to expand and "push" them forward. Here's the second difference between these two scenarios: the negative mass (again, according to Einstein's field equations) causes the space in front of the crew and their spaceship to shrink. Not only are they being pushed along at FTL speeds, but they also have the convenience of not having to travel as far. The negative mass stored in the crew's ship is like some kind of "space distortion machine," which also acts as an engine, which pushes the spaceship forward by expanding the space behind it with the convenience of also shrinking the space in front of it. This method of space transportation is called an Alcubierre drive.
The Einstein Field Equations (EFEs) are analogous to Maxwell's equations in the respect that both allow you to "work backwards," so to speak. Typically, Maxwell's equations have had been used to determine exactly what the electromagnetic field looks like given the distribution and dynamics of charge and current. But they can also be used the other way around. If you tell me what the electromagnetic field looks like, I can use Maxwell's equations to determine what the distribution of charged particles are doing. The same exact thing is also true for the EFEs. Historically, most applications of the EFEs went like this: given the distribution of mass and energy (which is captured by the stress-energy tensor \(T^{uv}\)), I can plug \(T^{uv}\) into the EFEs to determine the curvature and dynamics of spacetime (which is captured by the metric \(g^{uv}\)). In other words, plug \(T^{uv}\) into the EFEs, pull the mathematical crank, and you'll get the metric \(g^{uv}\) as an output.
But in 1994, proffesor Michael Alcubberie used the EFEs "backwards," so to speak. He started out by positing a metric (which described a locally flat spacetime with space behind it expanding and space infront of it contracting) and then he used the EFEs to determine what \(T^{uv}\) (which represents the distribution of mass and energy) could have manifested the particular spacetime curvature and geometry represented by \(g^{uv}\). What he found is that negative mass as massive as a star had to be distributed in the shape of a ring in order for the EFEs to "crank out" this particular \(g^{uv}\). So, despite the fact that Alcubberie proved the the Alcubberie metric is a valid solution to the EFEs and is therefore theoretically possible, a few problems arise: the first is that we've never seen negative mass. This problem might be eliminated since we have found naturally occuring negative energy (namely, dark energy or vacuum energy) and we've also created negative energy in the labaratory. The second problem is the impracticallity of assembling an amount of energy \(E\) which is equivalent to the mass \(M\) (according to \(E=mc^2\)) of a star. Research in 2011 involved changing the thickness of the ring to lower the mass requirement down to about the mass of Jupiter and further research done by the NASA scientist Harrold White has lowered that mass requirement down even further to about just the mass of a Voyager spacecraft.
Sources: PBS Space Time