Here's the best way I can sum up this story: Yes, some NASA scientists are working on a design for a warp drive. No, that doesn't mean warp drives are real.
Warp drives — as a purely theoretical thing and/or science-fiction plot device — involve manipulating space-time to allow a spaceship to go faster than the speed of light. It's basically loophole that would allow you to get around those pesky laws of physics. Swiss bank account:taxes::Warp drives:speed of light. You get the picture.
Harold White of NASA's Johnson Space Center is currently leading an effort to design a warp drive space ship. But, as Amy Teitel explains in a story for Vice's Motherboard, the fact that this is happening does not necessarily mean a real working warp drive is possible. It's more about the fact that NASA is partly in the business of letting really smart people try things that are kind of crazy and unlikely, if they can back up the idea with a reasonably plausible hypothesis. Speculative research is a thing that happens.
The problem is that breaking the light barrier isn't at all like breaking the sound barrier. The sound barrier–properly, the aerodynamic effects of pressure waves interacting with a body as it approaches the speed of sound–was broken with a cleverly engineered aircraft and an at-the-time state of the art rocket engine.
Bell's X-1 was, importantly, a physical aircraft made of matter, not made of sound. But the atoms and molecules that make up all matter are connected by electromagnetic fields, and that's the same stuff that light is made of. So when it comes to breaking the light barrier, it's like breaking through light with light (sort of… ask Brian Greene). As NASA poses the question, "How can an object travel faster than that which links its atoms?" It's a very different matter.
Another issue special relativity brings up is the light speed barrier. Moving takes energy, and the faster you move the more energy you use. So, theoretically, to move at the cosmic speed limit of light you need an infinite amount of energy. That's a distinct barrier if there ever was one.