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We now know that there are (at least) three flavours (types) of neutrinos: the electron-neutrino, the muon-neutrino and the tau-neutrino (this last one has not been observed yet, but its existence is inferred by analogy) and their anti-particles. We do not know if neutrinos have mass since all attempts to measure their mass have failed (see neutrino experiments).
However, if neutrinos actually have mass, it does not necessarily mean that the electron neutrino has a fixed mass, the muon-neutrino has another fixed mass and the tau-neutrino yet another. It is possible that an electron-neutrino, for example, is a composite particle made up of different massive neutrino states. This might sound like a weird idea, but actually this is exactly how the different types of quarks (the constituents of all hadrons such as nucleons and other baryons or mesons) operate amongst themselves. In fact, the quarks that suffer decays are a mixed state of the quarks that have a definite mass. This property is called mixing, so it is thought that if neutrinos have mass, they too could be in a "mixed mass state".
For simplicity, we could assume that for example the electron-neutrino is made up of two mass states (which we could call 1 and 2), so if an electron-neutrino is created in some interaction (for example, in the sun) then as it travels, each of the mass states travels with a different speed. This means that the electron-neutrino travelling through space is no longer a "pure" electron-neutrino but might be partly electron-neutrino and partly muon-neutrino. As the neutrino continues to travel, the proportion of each vary with distance, so it is said that neutrinos oscillate from one state to another. If we set-up a detector along its path, it would then be possible to observe not only the interactions of the electron-neutrino but the interactions of the other component (muon-neutrino in this example). If we saw muon-neutrinos where we would only expect electron-neutrinos we would observe the phenomenon of neutrino oscillations (appearance experiment), but it could also manifest itself if we saw that some of the original neutrinos were not there any more (disappearance experiment). As one can see, it is absolutely necessary that for this property to be visible that neutrinos must have more than one mass state (that is, neutrinos must be massive and the masses of each of the mass states must be different ). The proportion in which the two mass states can mix inside each neutrino flavour is called the mixing angle and is not known. If neutrino oscillations could be observed, this would be one of the parameters (with the mass difference) that could be determined.
There are a large number of experiments trying to observe neutrino oscillations. Some rely on man-made sources like nuclear reactors or accelerators and others rely on "natural" sources such as solar neutrinos or neutrinos from cosmic-rays (otherwise known as atmospheric neutrinos). All of these nutrino oscillation experiments are complementary because they involve neutrinos of different energies travelling over differnt distances. Since we do not know what the mixing angle and the mass difference is between the neutrino species we need to try and cover as much of our "parameter" space as possible to be able to discover oscillations in the future.
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Another motivation for studying whether neutrinos have mass or not is to try and determine whether neutrinos form part of the famous dark matter problem. Astrophysicists have been observing for some time that the rotational speed of galaxies is not what they would expect if the total mass of the visible stars made up the total mass of the galaxy. The rotational velocity of the stars at the edges of galaxies is much larger than what would be expected if most of the mass of the galaxy was concentrated close to its centre (galactic rotational curves). This implies that there must be an "invisible" form of mass that goes out to the edges of galaxies forming a halo of dark matter. Calculations of the percentage of dark matter vary, but it is believed that the visible matter makes up only between 1% and 10% of the total mass of the universe. The amount of dark matter is a crucial parameter to know if we want to determine what is the future fate of the universe. If the mass of the universe is above a certain critical mass, the current expansion would eventually halt and the universe would commence an implosion into itself, resulting in a "big crunch" at some time in the distant future. If the universe is below this critical mass, then the universe would continue to expand for ever and if it was at exactly the critical mass then it would also continue to expand but at a continuously slower rate.
There are a number of candidates for this dark matter: some are astronomical objects like MACHOs (Massive Astronomical Compact Halo Objects) which are low mass stars like brown dwarves or large planets similar to Jupiter or black holes with masses of less than a solar mass, or sub-atomic particles that have yet to be discovered (like Weakly Interacting Massive Particles or WIMPS, and axions) or neutrinos with a mass of the order of 1-30 eV. It is worth noting that MACHOs have already been discovered by the MACHO and EROS collaborations by the technique of gravitational lensing, in which the image of a distant object is amplified by a massive object in the light path between the earth and the far-away object, but the number of these objects is not sufficient to explain the whole dark matter story. There are many other experiments that are searching for dark matter and links to these experiments can be found through the UK dark matter search site .
It is well known that there is a cosmic microwave background that permeates the universe with an average temperature of 2.726 K. The observed universe shows rather clumpy features (large voids and areas of the universe with clusters of galaxies) and the uniformity of the microwave background in the universe seemed at odds with this clumpy structure. The Cosmic Observatory Background Explorer satellite (COBE) was launched to search for ripples in the microwave background that would be compatible with the clumpiness of the observed universe. The discovery of these ripples was made in 1992, in which it was found that the temperature of the microwave background varied by differences of about one thousandth of a degree in different parts of the sky. This was a triumph for the Big Bang theory of the universe, since it verified that the origin of the microwave background was in effect the remnant radiation from that Big Bang after cooling for more than 10 billion years and that these ripples formed the density fluctuations needed to form the large scale structure of the universe. Models that explain these fluctuations include the dark matter, and the COBE data favours a model in which there is a 70% cold dark matter (objects like MACHOS, WIMPS and axions which travel at non-relativistic speeds) and a 30% hot dark matter (like neutrinos which are relativistic particles) component. This still leaves the possibility open that neutrinos could make up about 30% of the dark matter. A logical candidate could be the tau-neutrino which could possibly be the heaviest of the neutrinos (assuming a mass-heierchy amongst neutrinos). This is one of the main motivations in the search for muon to tau-neutrino oscillations at experiments like NOMAD and CHORUS.