No, NASA has not discovered warp drive. There is a huge amount of media attention (here, here, here, for examples of relatively mainstream media) being given to a claim that a NASA team has successfully tested a gadget called the EmDrive. The claim is that one can take a conical microwave resonator (picture the cavity that is your microwave oven, only shaped like a truncated cone rather than a rectangular box), fire up microwaves to drive the resonant modes, and the cone will experience a steady thrust in one direction (the direction of the fat end of the cavity). There are multiple alleged explanations for this, ranging from botched thinking about special relativity to really bizarre word-salad about virtual particles, the quantum vacuum, and "warp fields".
Let me explain why this is bad science, terrible science journalism, and highly problematic.
First, the science. Our theory of electricity and magnetism is arguably the best understood, most precisely tested theory we have, both in its classical limit (the limit relevant for your microwave oven) and in its quantum limit (the limit relevant for things like calculating the magnetic moment of the electron, something that we can do to more than 14 decimal places! According to that theory, a closed microwave resonator does not generate thrust (surprise surprise). Given over 100 years of tests of classical E&M, it's going to take more than one poorly documented experiment, not published, to convince scientists that something exotic is going on. Extraordinary claims require extraordinary evidence, and this just isn't it. Moreover, claims that exotic quantum vacuum effects or "warp fields" are somehow relevant here are just on their face absurd! The energy densities, the materials involved, none of this couples to exotic quantum vacuum physics any more than my microwave oven does. This is like arguing that by accelerating a simple dielectric like a piece of plastic, I should see electron-positron pair production and warped spacetime. It's nonsense.
What would it take to convince me? How about a thoroughly documented experiment done by someone with credibility in precision measurement, for a start.
As for science journalism: The number of outlets who uncritically pass along something like this is appalling. What's worse, they distort it even more - the third link up top not only claims that this is a reactionless drive, but that it will allow faster-than-light travel. What the hell? (Yes, I know that the Daily Fail is third-rate fish-wrap.) I fully expect to see a CNN story about this, and it will be terrible. This will propagate in the media for several days, and they will portray it as some underdog inventors showing that the Scientific Establishment is wrong, or they'll present this as an actual scientific controversy, when in fact the burden is all on the experimenters to show that their work (which flies in the face of decades of contrary evidence) is right. Hey, IFLS: You should be ashamed of yourselves for your coverage of this. Good grief - I thought part of your message was that people should, I don't know, think critically!
Why is this problematic? It's an issue because people don't trust science, in part because they end up reading uncritical bull like this and come away thinking that science is either a dodge, a scam, or entirely a matter of opinion, when in fact it's an approach to thinking critically about the world that has made possible all of modern technology and medicine.