In a nutshell: neutrinos are tiny, tiny, TIIIIINY particles, that fly through other stuff as if it weren't there. We've got millions of them flying through our bodies every second from various solar sources. Under our current understanding of the universe, though, these neutrinos have mass - they weigh something, albeit such a small value that we don't actually know for certain what it is - and thus shouldn't be able to travel faster than the speed of light in a vacuum (about 300,000 km/s). However, a neutrino detector in Italy, OPERA, have calculated that some neutrinos created at CERN and fired towards them have travelled faster than this speed of light, c. This is extremely interesting, as it shouldn't be possible, and if it is then it completely ruins most accepted theories about the universe, and the scientists have to start from scratch again.
All flavor-changing means is that neutrinos experience time in some fashion, and thus do not travel at c. The typical assumption is that they have traditional mass, but imaginary or negative mass isn't completely ruled out.
We wouldn't be starting from scratch. GPS units won't stop working just because we find something that appears to move at 299,792,466 m/s, and they will continue using GTR (general theory of relativity) metrics that declare c=299,792,458 meters per second, because that's what it is.
But it would mean that a central tenet of our understanding of reality no longer holds. Either causality is violatable, or there is a preferred frame - and no preferred frame can exist within our Universe, so our Universe would need to be a hypersphere or some construct with similar properties.
I should point out too that time travel is still not possible in the science-fiction sense of the word. Nothing can travel backwards in time, it would merely be that the neutrinos arrive somewhere before electromagnetic radiation from the event. The event would have still happened X time ago, but the neutrinos might arrive before the light from the event. Sort of like when the tide goes out before a tsunami, I guess - you haven't seen the tsunami yet, but you know its going to happen because the water has receded.
Well, not time travel in the sense that you can just hop in a space ship and go back in time to an arbitrary point. It's a causality violation, which requires some gymnastics to engage in.
An attempt at a simple explanation requires pointing out a postulate of Relativity: The speed of light is the same in every frame of reference.
That is, for example, no matter how fast you go, you will always see light moving at the same speed. It may become redshifted (if you are moving away from a light source) or blueshifted (if you are moving towards it), but if you time shadows, or bake marshmallows in a microwave, light will always be 299,792,458 meters per second.
The math to make this work is given in Special Relativity, typically described as Minkowski spacetime and only requires algebra to understand. If something is moving close to the speed of light with respect to you, that something does not seem to experience much time, from your perspective - we call this time dilation. This is why, if you leave at .86 of c to go to Alpha Centauri, you only experience ~two and a half years passing, though an observer on Earth will experience five.
Something very wrong goes on when something carries information faster than c, however (there are some caveats to this, which I will note). If something can move information faster than c from your perspective, there will exist a frame of reference - something moving fast enough with respect to you - in which that information-carrying particle travels backwards
in time. According to Relativity, all frames are equally valid - there is no preferred frame, and this is certainly true for everything up to and including the speed of light. Taking this past c means that, if Relativity still holds, you will incur causality violations.
This is easiest to show with ansibles - an 'instant' communication device, where one message arrives somewhere else instantaneously, from your frame of reference (because 'x times the speed of light' has no real meaning for x > 1, so instantaneous makes for easy math). The time dilation factor at .86 of c is 2, so if you and your friend part ways at that speed (it doesn't matter who accelerates the first time), and you start sending messages to each other, you can send one at 8 seconds, they get it at 4 seconds, you get their reply at 2 seconds, they get yours at 1 second...
Actual, true to form time travel requires finding or making a friend in the early Universe several billion light-years away with this method, or some similarly convoluted scheme. "Make an infinitely long cylinder and rotate it."
It's also possible that an FTL communication medium could instead expose a preferred frame - you can still perceive things as going back in time, but this turns out to be nothing more than an illusion. No preferred frame can exist within our Universe, however. It would have to be extradimensional or some property inherent in spacetime that is invisible to us as is.
There are several versions of FTL travel that do not violate relativity or permit causality violations. These are somewhat limited, however.
Matter falling into a black hole will fall in faster than light. This is hidden from any observer, however, so is not considered 'usable'.
Likewise, the expansion of space is also faster than light at the edge of the visible Universe. Likewise, this effect is hidden from any possible observer.
A science-fiction scenario is a fixed-endpoint wormhole. A relativistic craft flying by its endpoint also perceives the shape of the wormhole to be deformed (length contraction), preventing causality violation.
But any free-form message-sending - such as an FTL neutrino - would either violate relativity or permit a causality violation as above.
It's been noted that the velocity seems to be increasing with neutrino energy. As all models of tachyons involve increasing speed with decreasing energy, this reinforces the general consensus that there is a synchronization error of some sort here.