Short answers, pulled from the story.
Arthur Holly Compton discovered the Compton effect in 1923 while studying the scattering of X-rays by light elements at Washington University in St. Louis. The discovery earned him the Nobel Prize in Physics in 1927.
The Compton wavelength of the electron is 2.43 in the units Compton used, and it represents the maximum possible wavelength shift in a Compton scattering event. The minimum shift is zero and the maximum is twice the Compton wavelength.
Compton scattering is the most probable interaction of gamma rays and high-energy X-rays with atoms in living tissue. Because it dominates in the photon energy range between the photoelectric effect and pair production, it is central to how radiation is deposited in the body during radiation therapy.
Inverse Compton scattering occurs when low-energy photons gain energy by colliding with high-energy electrons, reversing the usual energy flow. It is observed around accreting black holes, where it is thought to produce the power-law X-ray spectra seen between roughly 0.2 and 10 keV, and in galaxy clusters via the Sunyaev-Zel'dovich effect.
The Sunyaev-Zel'dovich effect is the result of cosmic microwave background photons being scattered to higher energies by hot electrons in the gas surrounding galaxy clusters, an instance of inverse Compton scattering. Observations of this effect provide a nearly redshift-independent method for detecting galaxy clusters.
Compton's experiment demonstrated that light cannot be explained purely as a wave phenomenon. It convinced physicists that light behaves as a stream of particle-like objects called photons, whose energy is proportional to the wave's frequency and which carry momentum as well as quantized energy.