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The Wikipedia article on beta decay explains it very well. As is the case with many great discoveries, it's useful to go back to the beginning, and try to see things through the eyes of those who first saw the evidence:

The study of beta decay provided the first physical evidence for the existence of the neutrino. In both alpha and gamma decay, the resulting alpha or gamma particle has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. However, the kinetic energy distribution, or spectrum, of beta particles measured by Lise Meitner and Otto Hahn in 1911 and by Jean Danysz in 1913 showed multiple lines on a diffuse background. These measurements offered the first hint that beta particles have a continuous spectrum.
In 1914, James Chadwick used a magnetic spectrometer with one of Hans Geiger's new counters to make more accurate measurements which showed that the spectrum was continuous. The distribution of beta particle energies was in apparent contradiction to the law of conservation of energy. If beta decay were simply electron emission as assumed at the time, then the energy of the emitted electron should have a particular, well-defined value.
For beta decay, however, the observed broad distribution of energies suggested that energy is lost in the beta decay process. This spectrum was puzzling for many years.
From 1920–1927, Charles Drummond Ellis (along with Chadwick and colleagues) further established that the beta decay spectrum is continuous. In 1933, Ellis and Nevill Mott obtained strong evidence that the beta spectrum has an effective upper bound in energy. Niels Bohr had suggested that the beta spectrum could be explained if conservation of energy was true only in a statistical sense, thus this principle might be violated in any given decay. However, the upper bound in beta energies determined by Ellis and Mott ruled out that notion. Now, the problem of how to account for the variability of energy in known beta decay products, as well as for conservation of momentum and angular momentum in the process, became acute.
In a famous letter written in 1930, Wolfgang Pauli attempted to resolve the beta-particle energy conundrum by suggesting that, in addition to electrons and protons, atomic nuclei also contained an extremely light neutral particle, which he called the neutron. He suggested that this "neutron" was also emitted during beta decay (thus accounting for the known missing energy, momentum, and angular momentum), but it had simply not yet been observed. In 1931, Enrico Fermi renamed Pauli's "neutron" the "neutrino" ('little neutral one' in Italian). In 1933, Fermi published his landmark theory for beta decay, where he applied the principles of quantum mechanics to matter particles, supposing that they can be created and annihilated, just as the light quanta in atomic transitions. Thus, according to Fermi, neutrinos are created in the beta-decay process, rather than contained in the nucleus; the same happens to electrons. The neutrino interaction with matter was so weak that detecting it proved a severe experimental challenge. Further indirect evidence of the existence of the neutrino was obtained by observing the recoil of nuclei that emitted such a particle after absorbing an electron. Neutrinos were finally detected directly in 1956 by Clyde Cowan and Frederick Reines in the Cowan–Reines neutrino experiment. The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.
Source: Wikipedia — Beta Decay

The significance of the "continuous spectrum" (meaning that the velocity of individual electrons emitted by beta decay from elements like radium and thorium varies over a wide range, with a few peaks) is key. It implies that momentum is not conserved in the subatomic processes that cause the electron to be formed and then to be kicked away from the site of its creation, which (though this wasn't known at the time) was a neutron or a proton, each of which weighs almost 2000 times as much as the emitted electron.

The conservation of momentum is an absolutely solid, immutable law of physics.

In order to explain the continuous spectrum of beta particle energy, there would either (1) have to be a violation of the conservation of momentum, or (2) some other particle would have to be emitted.

Wolfgang Pauli was the first to see this possibility, nearly twenty years after the experimental evidence had been observed and published. He called his hypothetical particle "neutron"; the particle we know today as the neutron wasn't discovered until 1931, so no one had used that name yet. It was Enrico Fermi who resolved that conflict by proposing to call Pauli's particle the "neutrino," meaning "little neutral one" in colloquial Italian.

Pauli didn't get his Nobel prize for the discovery of the neutrino, he got it for what has come to be known as the "Pauli exclusion principle," which is perhaps even more momentous a discovery than the neutrino... which is really saying something.

The first Nobel prize awarded in connection to the study of neutrinos was awarded to Clyde L. Cowan and Frederick Reines in 1956; they were the first to prove the existence of neutrinos by direct detection, more than 40 years after the first indirect evidence had been provided by Lise Meitner and Otto Hahn. Several additional Nobel prizes have been awarded to neutrino researchers since.