In principle, to show that a particle has mass, you just need to show that it travels at less than the speed of light. However, most neutrinos are produced traveling at almost the speed of light, and they interact extremely weakly with all forms of matter, so they are very hard to slow down. Instead, most searches for neutrino mass rely on a quantum-mechanical interference effect called neutrino oscillation. If neutrinos have mass, and if the neutrinos produced by the weak interactions do not have definite mass, but are quantum-mechanical mixtures of states with different mass, then the composition of a beam of neutrinos will oscillate, or change flavor back and forth with distance from the source. For example, it could oscillate from electron neutrino to muon neutrino.
Neutrinos pass through all forms of matter almost without interacting.
Sunlight comes just from the surface of the sun, but neutrinos are produced deep in its core, where the fusion reactions that power the sun take place. It was thought that the neutrinos should escape unfettered, but the first experiment to observe neutrinos from the sun, in the Homestake gold mine in South Dakota, found in the 1970s only about one third of the expected number. An elegant explanation of the solar neutrino problem has been provided by Mikheyev, Smirnov and Wolfenstein. These authors pointed out that neutrino oscillations in the sun, due to neutrino mass, can be modulated by the different interactions of electron and muon neutrinos with the electrons in the sun. The mass turns out to be a few thousandths of an electron volt.