Physicists in the 1920s had serious problems with -decay. If some radioactive material is decaying and each nucleus produces a single electron, conservation of energy and momentum requires that all the electrons should have the same energy. However, it was observed that the electrons had a continuous spectrum of energies.
This discovery led the great Danish physicist Niels Bohr to wonder if energy and momentum conservation did not hold for atomic processes. However, Wolfgang Pauli didn't like the idea of giving up on two principles that had served physics so well for so long. Instead, he postulated in 1930 the existence of a neutral particle that was also emitted during the decay. This particle would then carry away some of the energy from the decay, so the electron could have a variety of energies (hence giving rise to the observed continuous spectrum).
Detecting the neutrino turned out to be very difficult as they interact only extremely weakly with matter - most neutrinos would pass through a light-year of lead without being absorbed. You need a huge number of them in order to see anything. In fact, it was the antineutrino that was eventually detected first, by Frederick Reines and Clyde Cowan in 1956. The antineutrinos were from a nuclear reactor in North Carolina. The products of the fission of uranium tend to undergo -decay, and so nuclear reactors produce extremely large numbers of antineutrinos. Many trillions of antineutrinos entered Reines and Cowan's detector each second, yet in 200 hours they only detected a few hundred of them.