Compton scattering is an interaction between a photon and a charged particle (e.g., electron) where energy is passed from the photon to the particle, increasing the wavelength (decreases the frequency and energy) of the photon and changing the velocity of the particle, adding to its kinetic energy. Both the photon and the particle change direction. The change in observed spectrum is called Comptonization.
Inverse Compton scattering is the inverse: kinetic energy of the electron is passed to the photon, decreasing its wavelength. This is also called Compton cooling as it reduces the speed of electrons. In observing a cloud or body that produces such scattering, the Compton Y-parameter (sometimes just referred to as the Y-parameter or the Compton parameter) is the number of scatterings times the energy gained per scatter instance.
Gamma rays from the fusion at the center of a star are Compton-scattered many times in the surrounding plasma, producing the stellar spectra in the visible range.
Inverse Compton scattering of CMB photons passing through a galaxy cluster causes the Sunyaev-Zel'dovich effect.
As Compton scattering results in absorption and re-emission, the spectrum of electromagnetic radiation passing through a cloud of charged particles is changed, e.g., away from being pure black body. It is theorized to raise the energy levels of X-ray photons from black hole's accretion disks above what would be expected from a black-body spectrum.
In discussion of radiative transfer, Compton scattering and the lower-energy analog, Thomson scattering are referred to as electron scattering, because an electron's direction is changed, i.e., it is scattered, and/or because it causes a photon's direction to change, i.e., it scatters the photon.