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A neutrino is a type of subatomic particle. Specifically, it is a lepton. Neutrinos come in three flavors — electron neutrinos, muon neutrinos, and tau neutrinos (which correspond to the electron, muon, and tau, respectively). Neutrinos are electrically neutral, don't interact through the strong force, but do interact through the weak force. For many years, they were thought to be massless, but recent observations have shown that they actually have a small mass.
History and discovery of the neutrino
The neutrino (specifically, the electron neutrino) was first proposed by Wolfgang Pauli to resolve the apparent non-conservation of energy and momentum in nuclear beta-decay. Essentially, a neutron would decay to a proton, and a high energy electron would be emitted by the nucleus. This electron would have a certain, fixed energy. However, there was a continuous spectrum of electron energies and the energy was always less than what was expected. Pauli proposed that perhaps some of the extra energy was being carried off by an unseen particle — the neutrino. (Pauli actually called it a neutron. This meant that there were two types of neutrons — the nucleon and Pauli's particle. Enrico Fermi resolved this by calling Pauli's particle a neutrino, which means little neutral one in Italian.)
A while later, the neutrino that Pauli proposed was detected. (The one Pauli proposed is the electron neutrino. The muon and tau neutrinos have also been observed.)
Neutrinos are electrically neutral so they do not interact through the electromagnetic force. They do not have color charge so they do not interact through the strong force. They do however, interact through the weak force.
For a long time, neutrinos were thought to be massless. However, recent observations have shown that neutrinos do, in fact, have a small mass. Specifically, scientists observed neutrino mixing, which can only happen if neutrinos have mass.
Lepton number is a property of particles. Leptons have lepton number +1, and antileptons have lepton number -1, and everybody else has lepton number 0. In the Standard Model, lepton number is always conserved. However, there is no particularly deep reason for this. Such symmetries are dubbed accidental symmetries. In Grand Unification Theories (GUTs), lepton number and baryon number (which is the quark analog of lepton number and is also an accidental symmetry) are not conserved.