The Experimental Result
In a study by the OPERA group of over 160 scientists, led by Antonio Ereditato, neutrinos were sent from CERN in Generva, Switzerland to Gran Sasso in Italy, a total of 731 km (454 miles). The time taken was approximately 2.44 milliseconds, 0.0025% faster than the speed of light.
Modern physics has a wide range of subatomic particles which decay often within a fraction of a second into other particles. Neutrinos are also subatomic particles which emerge from these decays. They are very light weight particles.
Originally thought to be massless, the particles are now known to have a "rest" mass of several eV (electron volts) compared with 500 million eV (MeV) for the electron. Because they are so light weight, and carry no charge, they interact extremely weakly with other matter.
To filter out particles emitted from the sun and in cosmic rays, neutrino experiments are carried out deep below ground level.
Why the Result Could Be So Significant
According to the special theory of relativity developed by Einstein and Lorenz at the turn of the 20th century, as objects approach the speed of light they become progressively heavier. Physics thus refers to the "rest" mass of particles: their mass at zero velocity. A particle which has a non-zero rest mass cannot travel at the speed of light according to the theory since its mass would be infinite at that speed.
It's therefore not possible to accelerate particles above the speed of light, at least according to the theory. By comparison, particles which have zero rest mass can only travel at the speed of light, and this would include light itself (photons).
Time Travel
Traveling close to the speed of light is bizarre by our every day experiences. It is theoretically possible to travel into the future by traveling close to the speed of light. According to the special theory of relativity, your own clock is slowed down relative to the original clock ("time stands still on a beam of light").
So you "simply" travel off in one direction for a while at high speed, then turn around and come back. You then will have aged less than people on Earth. Of course the energies required to do this are way beyond our capabilities, except for a small fraction of a second of "time travel".
Time travel into the past is not allowed according to the special theory of relativity. If the new results are confirmed, though, this restriction might be lifted. Nonetheless, before considering this option, physicists will be looking for other explanations for the results.
Seeking an Explanation
The OPERA group analyzed their results in great depth but could not see any reason for errors. It is conventional in physics experiments to estimate the errors in the measurements. These are of two kinds, systematic and random.
The first error is an unavoidable error, such as the precision to which the distance between the two points is known and the precision to which the neutrinos could be timed. Random errors could have any sign plus or minus and are inherent in any measuring system. These can be reduced to the bare minimum by taking more measurements.
In the case of the OPERA group study, over 16000 neutrinos were tracked and the distribution of the timings was investigated. When the errors were analyzed, it was found that the mean time of travel was faster than light by 60.7 +/- 6.9 (stat.) +/- 7.4 (sys.) nanoseconds, where stat. is the random error and sys. is the systematic error. This difference is considered small but highly statistically significant.
A Theoretical Explanation
If the results are confirmed, physicists will also be considering other explanations for the neutrinos traveling faster than the speed of light. As discussed by Prof. Brian Cox for the BBC, one of these is the possibility of other dimension than the three spatial dimensions we observe, and that the neutrinos are not breaking the light barrier, but taking a "Shortcut through another dimension".
The Scientific Method
By posting the OPERA results on the internet, all knowledgeable scientists will get the opportunity to see if any mistakes were made. Assuming that none are found, other experimentalists will need to be able to repeat the results before they will be accepted by the scientific community. This may take as much as 5 to 10 years.
A similar strategy was adopted by the scientific community in the investigation of "cold fusion". Fusion is a process of production of nuclear power by the fusing of atoms together. It is the process that powers the sun. However this takes place at extremely high temperatures, 10 million degrees C or more, so the occurrence of fusion in the laboratory at room temperature was questioned.
Subsequently, no other laboratory could produce cold fusion and so the original results were discounted. It is this method of rigorous scrutiny of data and theory which marks the scientific method as arguably one of mankind's great philosophical successes. It is precisely this method which has led to the development and strengthening of other diverse theories such as evolution and climate change.