Why do neutrino oscillations imply nonzero neutrino masses? Neutrinos can pass from one family to another (that is, change in flavor) in a process known as neutrino oscillation. The oscillation between the different families occurs randomly, and the likelihood of change seems to be higher in a material medium than in a vacuum. 
Why does the oscillation of neutrinos directly imply that they must have a nonzero mass, since the passage of one flavor to another can only occur in massive particles? And What are the cosmological implications of these oscillations?
 A: The 3 neutrino families ($e$, $\mu$, $\tau$) are usually called neutrino flavors. 
Neutrino flavor oscillation requires that the mass eigenstates of neutrinos are not equal and that the mass eigenstate is also not a flavor eigenstate.  Since a neutrino is always produced in a flavor eigenstate (i.e. associated with an $e$, $\mu$, $\tau$), this flavor eigenstate wave function will be a mixture of the 3 mass eigenstates such that at the time of production it is a pure flavor eigenstate.  However as the neutrino wave function propagates, the 3 mass eigenstates will effectively move at different speeds so that at the point in space where the propagating neutrino interacts with the measuring apparatus, it will be a different mixture of flavor eigenstates.  Thus the possibility of flavor oscillation requires that the masses of the mass eigenstates are not equal.
That is why the flavor oscillation experiments always measure differences in masses (squared) but not absolute masses.  So since oscillations among all three flavors have been observed, there must be 3 different mass eigenstates with different masses.  However, the flavor oscillation experiments would allow the lightest mass eigenstate to have zero mass but would require that at least 2 mass eigenstates have masses that are non-zero and are not equal.
It is commonly assumed that all 3 masses are non-zero and not equal.
Note, for example, that you cannot talk about the mass of the electron neutrino since the flavor eigenstate of an electron neutrino will be a mixture of the 3 different mass eigenstates.
A: Assuming relativity as we know and love it.
Massless particles move at the speed of light.
At the speed of light particles experience no time, and therefore the evolution of the wave function from a pure flavor state to a mixed state which is proportional $U\exp(-iEt)U^{*}$ where $E$ is the energy of the particular mass state, $t$ is the elapsed time, and $U$ is the unitary mixing matrix does not proceed.
Because neutrinos are unambiguously observed to mix, they must experience time, so they must not be massless. QED.
A: Because neutrino's flavour eigenstates are superpositions of mass eigentates, a zero-mass eigenstate is not in contradiction with observation since the neutrino of a any given flavour would experience mass and time flowing due to mass acquired from the superposition of two massive and one massless states. 
Because probability of transitions depends only on mass differences, like potential differences in electricity, we may not be able to decide whether or not we have a gauge invariance (setting the zero mass or zero volt is arbitrary) or whether all eigenstates are really massive until we devise another way to weigh the neutrinos...
