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Let's break the question down:
1. What is a neutrino?
In quantum physics, a neutrino is an elementary particle, one of the fundamental building blocks of the universe. It belongs to the "lepton" category of particles, the same as the electron. But contrary to the electron, the neutrino has no electric charge, and virtually no mass. While the electron interacts with other particles (such as protons) by means of the electromagnetic force, the gravitational force, and a less well-known force called "weak nuclear force", the neutrino only interacts with other particles through this last one. That means neutrinos can travel through any kind of matter virtually unimpeded. In fact, there are hundreds of billions of neutrinos passing through your body every second of the day.
2. Can neutrinos travel faster than light?
Einstein's special theory of relativity, one of the cornerstones of contemporary physics, states than nothing can travel faster than the speed of light in the vacuum. This theory has been tested in many different ways over the years and has always been confirmed. In 2011 a group of physicists did an experiment in which they shot a stream of neutrinos from a laboratory in Geneva, Switzerland, toward a neutrino detector buried deep inside a mountain in Italy. When the physicists measured the time it took for the neutrinos to travel from their source to the detector, they found that the calculated speed was 60 milliseconds faster than the speed of light, which surprised the whole physics community. However, in February 2012 these same scientists discovered that there were flaws in the equipment used to measure the neutrinos' speed, and that the original measure could be wrong. Physicists are still working on the problem, and are planning to conduct new experiments to determine the real speed.
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