Neutron scattering is the scattering of neutrons by particles. The phenomena is used in probing astronomical bodies (planets, including Earth, and moons) such as remotely measuring water near the body's surface.
The scattering, I gather, is largely due to diffraction, given the neutron's de Broglie wavelength. The scattering happens naturally in environments that include free neutrons and is used in scientific lab instruments using artificially-produced free neutrons in a similar manner to the use of electron scattering or probing with electromagnetic radiation. It offers an alternate view of atomic/sub-atomic matter because neutrons (mostly) don't interact with electrons, but rather the nucleus, and the smaller the nucleus, the more chance of scattering. As such, neutrons are most affected by hydrogen nuclei and among the other elements, are highly affected by carbon and oxygen.
Some reactions that produce neutrons (e.g., collisions between high-energy particles and nuclei) produce neutrons with a high kinetic energy (KE), and subsequent scattering slows them. By measuring the kinetic energy of neutrons from a known source (neutron spectroscopy), the elements through which the neutron has passed can be determined. This is used by lab equipment, but can also be used with neutrons generated by cosmic rays that have penetrated the surface a bit, producing free, high-KE neutrons, that have the opportunity to be scattered by the material around and above them, after which some scattered and non-scattered neutrons move upward out of the surface (leakage neutrons). This allows neutron spectrometers above the ground to detect hydrogen, and thus water (or methane) at or below the surface. This is used, for example, by orbiting Mars probes to map characteristics at and just below the its surface.