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Neutrino oscillations -- the quantum mechanical transformation of one neutrino flavor into another during propagation -- provide direct evidence that neutrinos have nonzero masses, which is the first confirmed physics beyond the minimal Standard Model. The oscillation probability depends on the mass-squared differences Delta m2, the mixing angles, and the baseline-to-energy ratio L/E, and the phenomenon has been observed in solar, atmospheric, reactor, and accelerator neutrinos.
Neutrino oscillations are the first and so far only confirmed phenomenon requiring physics beyond the minimal Standard Model. The Standard Model as originally formulated contains only left-handed neutrinos with zero mass (no right-handed neutrino fields, no Yukawa couplings, no mass terms). The discovery that neutrinos oscillate between flavors -- implying they have nonzero masses and mix -- was recognized with the 2015 Nobel Prize (Kajita and McDonald, for Super-Kamiokande and SNO).
The oscillation formalism is analogous to quark mixing but involves the PMNS matrix (Pontecorvo-Maki-Nakagawa-Sakata) relating the three flavor eigenstates (nu_e, nu_mu, nu_tau) to the three mass eigenstates (nu_1, nu_2, nu_3). The oscillation probability in vacuum for two flavors is P(nu_alpha -> nu_beta) = sin^2(2*theta) * sin^2(Delta m2 * L / 4E). The full three-flavor case involves three mixing angles (theta_12, theta_13, theta_23), one CP-violating phase (delta_CP), and two mass-squared differences. All three angles have been measured: theta_12 ~ 34 degrees (solar, large), theta_23 ~ 49 degrees (atmospheric, near-maximal), theta_13 ~ 8.5 degrees (reactor, small but nonzero -- measured by Daya Bay, RENO, Double Chooz in 2012).
In matter, neutrino oscillations are modified by the MSW effect (Mikheyev-Smirnov-Wolfenstein): electron neutrinos experience an additional potential from coherent forward scattering on electrons (via W exchange), which modifies the effective mass-squared difference and mixing angle. In the Sun, this effect produces a resonant enhancement of oscillation that converts the majority of electron neutrinos to other flavors. The MSW effect is also what makes it possible to determine the neutrino mass ordering using long-baseline experiments or atmospheric neutrinos propagating through the Earth.
The major open questions in neutrino physics are: (1) the mass ordering -- is nu_3 the heaviest (normal) or lightest (inverted)? (2) the value of the CP phase delta_CP -- is there CP violation in the lepton sector, and if so, how much? (3) are neutrinos Dirac or Majorana particles -- do neutrinos have distinct antiparticles, or are they their own antiparticle? The first two will be addressed by DUNE, Hyper-Kamiokande, and JUNO in the coming decade. The third requires observing neutrinoless double beta decay, a process that violates lepton number and is possible only if neutrinos are Majorana fermions.