I'm going to tell you how, soon, humanity might measure the masses of neutrinos just by observing past events in the universe. I like this topic because it is one of the few situations in fundamental physics where a measurement of the greater universe might detect something about fundamental particles and/or their interactions, before we manage to measure it in a lab. Another example is the existence of dark matter; however the mass of dark matter will almost certainly be first measured in a lab. Perhaps with neutrinos it will go in the other direction?
What is a neutrino?
I guess that before telling you how to measure a neutrino's mass, it might be pertinent to tell you what a neutrino is and how we can know it has mass before we've measured that mass. Well...
When an atomic nucleus decays, the decay products we see are other nuclei, electrons and/or positrons. These visible products always carry less energy and momentum than the amount that the initial nucleus had. This suggests strongly that some unknown other particle is also being created in the decay and that we just can't see it. This hypothetical particle was dubbed the neutrino and when theories were developed for the force responsible for nuclear decays, the neutrino became an important part of them. And, eventually, neutrinos were detected directly. It took a while because neutrinos interact incredibly weakly, which means you need either a lot of neutrinos or a lot of transparent stuff for the neutrino to interact with (or both) before you will see them.
Initially, it was assumed that neutrinos are massless. They don't need to be massless, but for a long time there was no evidence that they did have mass, so the simplest assumption was that they didn't. There are three types of neutrinos: those emitted in interactions with electrons, those emitted in interactions with muons and those emitted in interactions with tau particles. If neutrinos were massless, then a neutrino emitted as an electron neutrino would always remain an electron neutrino. Similarly, a muon neutrino would always remain a muon neutrino. However, if neutrinos do have mass, then a neutrino emitted in an interaction with an electron will actually travel as a superposition of an electron neutrino, muon neutrino and tau neutrino. The net result being that this neutrino could be detected as a different type of neutrino. Therefore, a smoking gun thing to look for when determining whether neutrinos have mass is this characteristic signal whereby one type of neutrino appears to oscillate into another type of neutrino.
This effect was then seen and seen and seen again. Neutrinos appear to have mass. From the perspective of particle physics this is a bit weird. Neutrinos must have really small masses and it is unclear why these masses are so small. Unfortunately, this mechanism of neutrino oscillations doesn't directly give the masses of the neutrinos. Although, it can be used to measure the differences of the masses of the neutrinos, thus setting lower bounds on the possible masses of the neutrinos.
What has this got to do with cosmology?