In recent weeks you might have seen news like this:
Faster than light particles found, claim scientists (Guardian)
Speed-of-light results under scrutiny at Cern (BBC)
Naughty 'Faster Than Light' Neutrinos a Reality? (Discovery)
And what reportedly happened was that neutrinos (ghostly elementary particles usually produced in weak nuclear decays) sent in a beam from the LHC in Switzerland to the OPERA detector in Italy 732km away arrived 60 nanoseconds faster than light would in the same distance.
Why this is significant
If this is true, that particles can travel faster than light, then Einstein would be wrong and the foundations of physics would be overturned.
This would imply, among other things, that
1. Time travel is possible (as effects can come before causes)
2. Space travel is possible (both trans- and inter-galactic space flight)
Caution is advised
Discounting the possible trivial errors (because those physicists have been getting this result for months already, and have been checking and re-checking for obvious mistakes right from the beginning, so it's unlikely that they missed any such simple blunders), we still shouldn't be too quick to jump to conclusions.
For one thing, other particles, when accelerated towards the speed of light, will actually increase in mass. For example, the protons accelerated at the LHC at maximum energy will be 7,000 times their original mass when at rest. But these neutrinos, even though they are not massless (like light itself) did not exhibit this kind of mass increase and subsequent radiation.
For example: http://blogs.scientificamerican.com/...-out-en-route/
For another, there are so many other less-obvious and subtle possibilities. Maybe there's something up with GPS coordinates at this level of precision required. Maybe there's some little-understood effects that impede radio wave transmission between the 2 facilities. Maybe there's some material in the ground that can accelerate the neutrinos in unexpected ways.
Of course, all of these could be wrong, but the point is, as Carl Sagan loved to say, "Extraordinary claims require extraordinary evidence".
Some comments I would add
However this turns out, we would definitely learn something new about the world around us. In a way science is exciting because anyone can and do make phenomenal discoveries anytime, so if everything turned out just as expected, it would also be disappointing because this means there is nothing new left to learn about our Universe.
And the results of this experiment is definitely very unexpected. For most of us in this field, our attention was focused exclusively on the LHC itself (including me), but while the stuff we expect like supersymmetric particles fail to show up, mind-boggling results showed up at an experiment not even designed to look for faster-than-light particles. So it's kinda like everyone looks at the stage but the magician shows up and does a brilliant trick behind the audience.
Unfortunately, it would be difficult to verify the results in the near future, because there are only 2 other existing facilities in the world that can handle this - the Japanese facility was damaged by the recent tsunami disaster, while the US facility is experiencing financial hardship and have shut down their main accelerator just this week.
So, we eagerly await both theoretical and experimental solutions.
NOTE: The original pre-publication paper describing the experiment can be found here: http://arxiv.org/abs/1109.4897