What is a Neutrino? A neutrino is one of the particles that make up other particles. It's part of the building blocks of atomic theory. Neutrinos carry no charge, so they are different than the electron. Since they have no charge they are able to pass through matter. Neutrinos also require very special detection mechanisms. Neutrinos also have mass.
This is why it is a big deal to have detected the speed of a neutrino at greater than the speed of light. Either the neutrino always has traveled faster than the speed of light or they were somehow able to accelerate to a speed greater than that of light, which requires infinite energy under our current model of physics. Since we're talking about a particle accelerator here it can be assumed that the collision created the neutrino, thus we know that it is impossible for infinite energy to have be entered into the system.
Now that we understand what is going on, what is at stake here? A particle that is able to accelerate to a speed faster than the speed of light completely shifts our understanding of subatomic particles. Actually, it obliterates it. We will have no clear understand of what is going on at these particle sizes.
Could this be the greatest finding in the 21st century? Yes. All physicists believe it would be. Are people just accepting these findings? No. There is a great deal of skepticism, and it's not just from the broader community. The scientists that are presenting the results are basically issuing a challenge to the scientific community to show that they are wrong! Based on their findings these results are in fact statistically significant.
Are other scientists going to test these results to verify it? Well, there are only two other places in the world that could have the capabilities to test it. Fermilab in Chicago and a Japanese lab that was damaged by the earth quake and tsunami. However, Fermilab's equipment isn't sensitive enough to detect the difference in the speeds. Basically, the speed difference is so small it is within the margin of error for the detection equipment.
What's this all mean to me though? Well, for us non-physicists life goes on as normal. We can't suddenly travel faster than light. However, this is a case of good science at work. We should seriously pay attention to what happens here. This type of science happens all the time at a smaller scale. For evolution this type of science is happening. Some extraordinary claim is made, which requires extraordinary data to support it, then is tested by other people. IF the claim withstands additional scrutiny the claim is accepted. In some cases where the claim is so extraordinary that the people making the claims don't really buy it, then it is the duty of the scientific community and the larger community to give them the support they need to determine the validity of the results.