The solar neutrino problem is a major discrepancy between our understanding of nuclear physics and our observations.
The sun is a natural nuclear fusion reactor, fusing hydrogen to helium. Our current understanding of physics is quite clear about what happens, four hydrogen nuclei (protons), with or without the help of catalysts are transformed into helium, neutrinos and energy. The energy is released as gamma rays, and kinetic energy of the particles, including the neutrinos.
Neutrinos were originally theorized to make up the energy and angular momentum difference when a neutron decays into a proton and an electron. Neutrinos were later proven to exist, which was difficult considering they have negligible rest mass (which means that their gravitational interaction is indetectably small), travel very close to the speed of light, and have no charge (this means that they do not interact through electromagnetic interaction, which is the only way to currently directly detect particles as small as neutrinos). They also don't interact through strong nuclear interaction. Fortunately, neutrinos do influence other matter through weak nuclear interaction. Large heavy water (water that has deuterium instead of hydrogen) tanks with arrays of photocells are usually used to detect neutrinos.
The detectors used to capture solar neutrinos are huge, usually deep underground to avoid noise from cosmic rays, and still have trouble detecting neutrinos. As the technology progressed, and bigger detectors were built, it became clearer that we just weren't getting as many neutrinos from the sun as our models of solar combustion predicted. So either our theories, which were by all accounts lovely constructions, were wrong, or our detectors were still too primitive or misunderstood, or neutrinos were changing on the fly. (Or the sun was going out, see the Science Fiction Wiki.)
Recently, experiments have shown that the neutrino has mass, that is, a non-zero rest mass. This would allow it to indeed change on the fly among forms of neutrino, a trick called oscillating, without changing the standard model of physics. This discovery seems to have solved the solar neutrino problem. The most notable are experiments are Super-Kamiokande (Super-K), in Japan, and the Sudbury Neutrino Observatory (SNO) in Canada.