Image of neutrino paths
This discovery was made by OPERA (Oscillating Project with Emulsion-tRacking Apparatus) and CERN (European Center for Nuclear Research). The CERN particle collider fired a beam of protons at a solid target, creating neutrinos which sped through 730 kilometers of earth to the OPERA neutrino detector.
The neutrinos travel through the earth
The neutrinos traveled this distance in 2.43 milliseconds, 60 nanoseconds faster than they should have. This is a small difference, but the experiment is supposed to have a margin of error of 10 nanoseconds, so this measurement should be valid.
OPERA neutrino detector
If this observation is accurate, it would unravel Einstein's theory of relativity. Over the years many theories have been based on Einstein's theory, so if it were to fall apart, most of what we know about physics would be wrong. I have heard some pretty extraordinary claims about what this all could mean. One of the more fantastic ones is that we might be able to violate causality by sending messages back in time. Another is that there is a warped 5th dimension in our universe where the speed of light is faster, and the particles just went through a pocket of this dimension. I don't believe any of this, it sounds more like bad science reporting than anything else. The final claim that I've heard, which makes infinitely more sense, is that the speed of a neutron will just become the maximum speed limit, not light.
Picture from the inside of the CERN particle collider
Nobody is assuming that this observation of the speed of neutrinos is correct. This is an extraordinary claim, so extraordinary that the researches who found it are assuming that there was some error in their measurements. They were incredibly rigorous in eliminating any errors that they could find however, so they published a paper to be scrutinized by the scientific community so that someone can figure out if anything went wrong. This experiment is to be verified in two other labs in the near future, no one is going to believe this claim until there is a lot of evidence to back it up. In science, we must remember that we don't really know anything, years of study and research can be overturned in a single night.
XKCD comic strip