The Alcubierre drive, or warp drive, works by expanding space behind the spacecraft and contracting space in front of the spacecraft without distorting spacetime in the bubble occupied by the spacecraft. This causes no acceleration g-forces, it is not limited by the light speed limitation since it is not propelled by momentum, and it does not cause any relativistic time effects such as time dilation or, when travelling faster than the speed of light, backward time travel.
There is a good reason not to redirect the whole warp drive page to Alcubierre drive: Alcubierres original paper assumed absurdly large requirements of negative energy. One possible, more efficient warp drive is described in manipulated spacetimes. There is also an experiment on the concept, see Warp drive experiment.
Ideas and possibilities
Short electromagnetic waves
Multiple scientific theories, including String Theory, independently predict that gravity and electromagnetism unify in higher dimensions. String theory and others predict that the higher dimensions are hidden because they are extremely small. If an electromagnetic wave is made short enough to fit into the hidden dimensions, it would manipulate space-time efficiently. This is the possibility directly predicted by string theory. The other, highly speculative possibilities have been moved to the talk page.
The engineering problem
Extremely short-waved photons are highly energetic, but it would only be necessary to produce a relatively small number of such photons, because they are very energetic. One way of producing such photons would be to focus a lot of laser energy onto very few atoms (ie a nanoscopic particle), heating them to a extremely hot plasma that would radiate a few ultra-short photons. The Alcubierre metric would require massive manipulation that, if triggered inside the craft, would tear the craft apart. One solution would be to eject nano-particles in controlled trajectories, and then fire lasers at them with perfect timing (ie coordinated by a atomic clock), so that multiple gravity-manipulating effects in a series are simultaneously triggered outside the craft (that is, all nano-particles are hit by laser beams at the same time, but the lasers fire at slightly different times to compensate the distance between the nano-particles). This would, by virtue of producing no discrete event horizon, avoid the problem about Hawking radiation.