What are mass eigenstates?

According to Wikipedia

Neutrino oscillation arises from a mixture between the flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three neutrino states of definite mass. Neutrinos are created in weak decays and reactions in their flavor eigenstates. As a neutrino propagates through space, the quantum mechanical phases of the three mass states advance at slightly different rates due to the slight differences in the neutrino masses. This results in a changing mixture of mass states as the neutrino travels, but a different mixture of mass states corresponds to a different mixture of flavor states. So a neutrino born as, say, an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance. Since the quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate as long as the quantum mechanical state maintains coherence.

What are these mass eigenstates?

We can write mass terms in the Lagrangian as $\phi^\dagger M \phi$ where $M$ is the mass-matrix. The problem with this is that $M$ may not be diagonal so we might have terms like $m^2 \phi_1 \phi_2$ in the Lagrangian, which is difficult to obtain physical meaning out of because such terms don't have a direct classical correspondence to mass terms in Newtonian Mechanics. So, we instead write $\phi^\dagger M \phi$ in a basis such that $M$ is diagonal and thus we have terms that only look like $m^2 \phi_1 \phi_1$ (and so on for the other fields). The $\phi$'s in this basis are known as the mass eigenstates.