Neutrinos are ghostly particles emitted as a by-product of various nuclear reactions, and were first proposed by Wolfgang Pauli and Enrico Fermi in an attempt to explain some of the most bizarre aspects of beta decay. Since the neutrino is electrically uncharged and has no (or perhaps negligible) mass, the means by which it can interact with ordinary matter and hence be detected are limited to events involving the weak nuclear force. There are only two known methods by which this can happen, inverse beta decay is one (neutral current reactions involving the Z particle are the only other way known to science as of this writing).
Basically inverse beta decay is just that, the reverse of regular beta decay or positron emission. It involves a proton absorbing an antineutrino, transforming into a neutron and a positron, or perhaps a neutron absorbing a neutrino, becoming a proton and an electron:
p + ν* -> n + e+
n + ν -> p + e-
(the antineutrino ν* should actually have an overline, but this is unfortunately the best I can do with HTML).
To be more precise, what happens in the first case is the antineutrino comes close enough to one of the up quarks inside the proton, the antineutrino emits a W - particle, turning into a positron in the process. The up quark absorbs the W particle and becomes a down quark as a result. In the second case the roles are reversed, neutrino emits a W + particle, turning a down quark into an up quark as a result. This is essentially the reverse process involved in normal beta decays. As one might imagine, the interaction has an extremely low probability of occurring, as it requires that the neutrino come close enough to a particle to exchange W particles, which are so heavy that their effective range is about 3 × 10-17 m (smaller than the size of a proton or neutron!), but again, these probabilities are definitely not zero.
The first experiments that confirmed the existence of neutrinos used this method, and most neutrino observatories (until the Sudbury Neutrino Observatory) exclusively used this method in studying these elusive particles. In 1953, Frederick Reines and Clyde Cowan used this reaction to produce the first conclusive proof of the existence of neutrinos by using the considerable neutrino flux coming from a running nuclear reactor. They dissolved cadmium chloride into a tank of water, and surrounded the tank with sensitive gamma ray detectors, and placed it near the nuclear reactor at Hanford (one of the first operational reactors in the world). The thinking was that the hydrogen nuclei (protons) that made up the water would wind up experiencing inverse beta decay from the neutrino flux coming from the nuclear reactor. If something like that happened, the positron released would quickly find an electron somewhere and two 0.51 MeV gamma rays would be emitted in the resulting pair annihilation. The neutron that the proton had become due to the inverse beta decay would eventually migrate through the water and within a few microseconds would get captured by one of the cadmium nuclei. The heavier cadmium nucleus would be unstable, and wind up emitting gamma rays in its radiocative decay, which would also be seen by the detectors. They wound up detecting too many spurious events that could be from cosmic rays as well as from inverse beta decay, so in 1955 they moved the setup to the new reactor at Savannah River, which was better shielded than the one at Hanford. There, they confirmed the neutrino hypothesis by noting the difference in number of inverse beta decay events with the reactor on and off, 21 years after Pauli and Fermi first formulated their hypothesis. Reines and Cowan had to wait twice as long themselves before their discovery was seriously recognized, as it was only in 1995, 40 years later, that Reines received a Nobel Prize in Physics for his attempts at confirming the neutrino hypothesis (Cowan having died in 1974).
Sources:
Arthur Beiser, Concepts of Modern Physics
Rubin H. Landau, Quantum Mechanics II
Dennis Silverman, "The First Detection of The Neutrino", http://www.ps.uci.edu/physics/news/nuexpt.html