Neutrinos are produced in a variety of decays and interactions. For example, a neutron decays into a proton, an electron, and an anti-neutrino. In fact, it was through careful observation of radioactive decays that physicists hypothesized the neutrino's existence.
For example, when a neutron decays into an electron and proton, the sum of the electron's and the proton's momenta is not equal to the original momentum of the neutron. Thus, there must be some other particle involved in this decay that accounts for the missing momentum: the neutrino.