For a long time, there was no
lower bound on the mass of
neutrinos. That is to say, they seemed
massless. But, some particle physicists got the bright idea of trying to detect oscillations in the
identity of the neutrinos.
That probably made no sense. Let me explain.
There are several particles which are similar enough that over time they can morph into each other. For example, the Kaon. It can either have charge, or none. Also, it can either have one mass, or another. But if you determine its mass, then its charge will oscillate. If you determine its charge, its mass will oscillate. This is related to the Heisenberg Uncertainty Principle. This process takes a well-defined amount of subjective time to occur (on average... these are quantum particles here).
Now, any particle that experiences the passage of time cannot be moving at the speed of light (see special relativity).Therefore, if neutrinos oscillate, then they must be moving slower than the speed of light. They can't be massless AND moving slower than the speed of light, because this contradicts E = γmc2 (*). We know that neutrinos have energy, so if they oscillate then they must have mass.
There are a variety of ways to detect neutrino oscillation. In general, you have to get a look at a population of neutrinos at one point in the population's life cycle, and then again at a later point. For example, the K2K experiment in Japan uses a particle accelerator to throw trillions of muon neutrinos at the Super Kamiokande ("Super K") neutrino detector. A few of those will interact in Super K's detection chamber, revealing what type they are. Excess neutrinos that are of the 'wrong' type (mainly, electron neutrinos) indicate that identity oscillations have occurred. The amount of this indicates how much mass they must have. At this point, the lower bound is fuzzy, but it is distinctly nonzero.
Another experiment is looking at the deficit of solar electron neutrinos -- it seems that as neutrinos stream out of the sun due to the fusion reactions there, some of the electron neutrinos are turning into muon and tau neutrinos. Neutrino oscillation explains the overwhelming lack of neutrinos observed in experiments that could only detect electron neutrinos, as compared to the theoretical predictions. That discrepancy used to shadow the solar physics community, but with neutrino oscillation, that worry is gone -- the neutrinos are all there.
* This is the more proper form of E = mc2. Now that we have access to constant fundamental masses like the masses of the various particles (something Einstein did not have available in 1905), it is most convenient to let only energy vary with speed.