What is a neutrino?
An extremely light, electrically neutral particle that interacts with matter so weakly that trillions pass through your body every second without effect — which is exactly what makes them both nearly undetectable and uniquely informative.
Why they were proposed. In the 1930s, beta decay appeared to violate conservation of energy — the emitted electron carried less than expected. Pauli proposed an undetected particle carrying the difference, calling it a desperate remedy and doubting it could ever be observed. It was detected experimentally in 1956.
Why they barely interact. Neutrinos respond only to the weak nuclear force and gravity — not to electromagnetism, having no charge, and not to the strong force. A neutrino could pass through a light-year of lead with a reasonable chance of emerging.
How they are detected anyway: enormous volumes of material, deep underground to shield from cosmic rays, watched continuously for the rare interaction. Detectors use tanks of water or liquid scintillator surrounded by thousands of light sensors, instrumented ice at the South Pole, and liquid argon chambers. You do not catch many; you watch a very great number of chances.
The discovery that changed physics. Neutrinos come in three types, and experiments found that solar neutrinos arriving at Earth were fewer than predicted. The resolution was neutrino oscillation — they change type in flight, which is only possible if they have mass. The Standard Model had assumed they were massless, so this was the first firm experimental evidence of physics beyond it, and it earned a Nobel Prize in 2015.
Why they are scientifically valuable. Because they pass through almost anything, they escape directly from places light cannot: the core of the Sun, the interior of a supernova, and the centre of distant galaxies. Neutrinos from supernova 1987A arrived hours before the light, having left the collapsing core immediately while photons took hours to escape.
What remains unknown: their absolute masses, why those masses are so tiny, and whether a neutrino is its own antiparticle.