Star Trek-style "warp" drive allowing faster-than-light (FTL) travel was shown to be theoretically possible in 1994, but the energy requirements were enormous: The entire mass-energy of a planet the size of Jupiter was required to power the drive, relegating the concept into the most distant future scenarios - timescales on the tens of thousands, hundreds of thousands, or millions of years, depending on one's level of optimism. But newly refined research by NASA tweaking the parameters of the theory finds that it's possible with a mass-energy no greater than a few hundred to a few thousand kilograms: A factor of 2.6 septillion (2.6 x 1024) reduction in energy requirements. And this is merely the second generation of the theory, so the possibilities for future improvements in theoretical practicality seem vast.
Real warp drive theory is called the Alcubierre drive after its theoretical pioneer, Mexican physicist Miguel Alcubierre. It functions by bending space around a spacecraft such that it creates a "bubble," and the bubble moves forward by expanding in front and contracting behind while the mass inside is stationary relative to space within the bubble - i.e., there is no violation of general relativity. As a result, the troublesome aspects of fast relativistic travel such as time dilation - i.e., the faster you go, the slower your time runs relative to the rest of the universe - would not apply. There would also be no g-forces from acceleration of the bubble, so as far as a ship inside is concerned, no intertia is involved. A ship capable of warp drive would theoretically be shaped like a ring surrounding an oblong payload/habitat system:
The colorful distortions in the above illustration may not have anything to do with how things would really look from inside a warp bubble - it may not look like anything, or it may just be a slight dicoloration and/or distortion of otherwise normal-looking star fields, although probably more pronounced ahead and behind than to the sides. Anyway, the new version of warp theory is able to achieve these massive energy savings by modeling a differently-shaped bubble: Instead of a sphere, the bubble is a donut - and apparently this requires a miniscule fraction of the energy. More advanced bubble shapes in future models may reduce the requirement further, although I doubt by similarly impressive margins.
Just as in Star Trek, the maximum theoretical speed using an Alcubierre drive is 10 times the speed of light - i.e., Warp 10 or 10c [Woops - apparently the meaning of Warp in Star Trek is not a direct multiple of lightspeed, so this analogy is invalid]. This is because the thickness of the bubble at 10c approaches the theoretical minimum size of cosmic structure, the Planck length. Ten times the speed of light is nothing to sneeze it, and it would make interstellar travel to the nearest stars feasible, but it's not as fast as it sounds: It would still take 5 months to reach Alpha Centauri, over a year to reach Epsilon Eridani, and 12,000 years to cross the Milky Way galaxy. And if we do achieve warp drive, it's more likely to be most practical at factors lower than 10c - the lower, the more likely.
In fact, the most probable and earliest application of an Alcubierre drive would be sub-light travel within the solar system. The challenges would still be tremendous compared to any other technology under development, but it would likely be vastly easier to achieve 0.1c or lower and yet still completely open the solar system to humanity. At 0.01c - a hundredth of lightspeed - it would take a day or two to reach Mars (as opposed to 9 months with current rockets, and 5 weeks with VASIMR), and a couple of months to reach Neptune, provided that acceleration and deceleration of the bubble could occur rapidly.
Now, we shouldn't overestimate how close this technology is, because despite the enormity of the theoretical advance, all that this means is that it would not violate the laws of physics to achieve FTL travel with a few thousand kilograms of mass-energy. The actual engineering - the specific how, rather than if - is by far the hardest part. After all, we have a pretty solid foundation of pursuing fusion reactor engineering, but humanity has not yet gotten there in generations of effort. I think we will eventually succeed, but exactly when is an open question. Nonetheless, this is a massive injection of optimism: I can imagine this being late 22nd century technology rather than 22,000th century technology. But even if it took a thousand years, I would count the advance huge.
For the moment, essentially all of the practicalities are wide-open questions: We don't know how to create the warp bubble without destroying its contents, how to bring it up to desired speed without destroying its contents, how to slow it down again without destroying its contents, and then how to "pop" it so the ship inside can reenter normal space without destroying said ship. And even if/when these questions are answered for sub-light speeds, that doesn't mean we would have the answers for FTL speeds, so this is very much an entire array of new frontiers to explore. One thing is always guaranteed: It is a lot more complicated than we imagine. But within that fact lays another - that the possibilities are more numerous and wondrous than we can imagine.
But here's the exciting part: It isn't just numbers on a whiteboard or in a computer model - practical experiments are being planned by the scientists involved in the new research to see if warp bubbles can be created on a microscopic level using lasers. The fact that such experiments can even be designed, let alone whether or not they succeed, is a giant leap for mankind. Progress in this field has just been one of the promising results to gain widespread attention via DARPA's 100 Year Starship initiative, which continues the organization's tradition of forward-thinking endeavors.
I should note, however, that FTL travel is not necessary for humanity to colonize other star systems - it would just take a lot longer, and involve a lot of intermediate steps with each new system before another one could be colonized from it. But FTL means the difference between colonizing the nearest habitable systems over 10,000 years vs. doing so over the next 500, and also drastically reduces the timline for conquest of our own solar system. Rather than taking 2,000 years to build flourishing civilizations in every suitable niche our solar system has to offer, it could happen over a couple of centuries like European settlement of the New World. So we probably won't be seeing warp drive ships in this century, but the first dim glow of a Star Trek dawn has peeked over the horizon. Even a massively disappointing outcome - e.g., if it's only practical for sub-light speeds - would make future history a lot more interesting, and develop a lot more quickly than otherwise.
Thu Sep 20, 2012 at 11:07 PM PT: Apparently I was incorrect about the meaning of warp in Star Trek - a warp factor does not refer to the number of times the speed of light, but to orders of magnitude or greater.
8:25 AM PT: A number of commenters have expressed skepticism about the viability of the theory to become technology. I don't see any basis to their objections, but I'm not an expert. The people responsible for the research discussed here are, however, so anyone who has the attitude that this is just "science fiction" or the like should really take it up with Dr. Harold White of NASA Johnson Space Center.