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 and without the help of catalysts are transformed into helium, neutrinos and energy. The energy is released as gamma rays and as 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 demonstrated to exist, but accomplishing this was difficult because they have negligible mass, travel very close to the speed of light, and have no electric charge nor magnetic moment (thus don't interact electromagnetically). They also don't interact through the strong nuclear force. Neutrinos do interact with other matter through the weak nuclear force (that is how they are produced). 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. 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. In various experiments, the number of detected neutrinos was between 1/3 and 1/2 of the predicted number. Therefore either our models of the sun were wrong or our models of neutrino behavior were wrong. This is known as the Solar neutrino problem.
The solutions based on the models of the sun being wrong were based on the premise that the temperature and pressure in the interior of the sun was not what we thought it was. For example, since neutrinos measure the amount of current nuclear fusion, it was suggested that the nuclear processes in the core of the sun might have temporarily shut down, and since it takes thousands of years for heat energy to move from the core to the surface of the sun, this would not immediately be apparent.
Solutions based on incorrect understanding of solar physics were rendered untenable by helioseismology which observes how waves propagate through the sun. Based on these observations it became possible to measure the interior temperatures of the sun and these agreed with the standard solar models.
Currently, the solar neutrino problem is believed to result from an inadequate understanding of the properties of neutrinos. According to the Standard Model of particle physics (prior to 1999), there are three different kinds of neutrinos:
The first evidence for neutrino oscillation came in 1998 from the Super-Kamiokande collaboration in Japan. It produced observations consistent with muon-neutrinos (produced in the upper atmosphere by cosmic rays) changing into tau-neutrinos. More direct evidence came in 2002 from the Sudbury Neutrino Observatory (SNO) in Canada. It detected all types of neutrinos coming from the sun, and was able to distinguish between electron-neutrinos and the other two flavors. After extensive statistical analysis, it was found that about 35% of the arriving solar neutrinos are electron-neutrinos, with the others being muon- or tau-neutrinos. The total number of detected neutrinos agrees quite well with the earlier predictions from nuclear physics based on the fusion reactions inside the sun.