"The short version", eh? Basically -- neutrinos are teeny tiny little particles that may or may not have mass. They are emitted by the sun and supernovas (stars that explode) and exist in vast quantities. This is why the question of their mass, however infinitismal, is important, because en masse their mass could be massive. If ya follow.
It's looking more and more like they do have mass, and could account for the "missing mass" (or dark matter) that makes the standard model work. Basically, some experiments have shown they do have mass, and they're trying to replicate the experiments, do variations, and generally get more data on whether they have mass and if so, how much.
I doubt E.G. will "sign up", but if you think of questions, let me know, and I'll run them past him at the dinner table and report back.