Raman scattering is inelastic scattering from molecules. The photon interacts with the molecule and changes the molecules vibrational, rotational or electron energy.
Rayleigh scattering is in the main elastic scattering from small particles whose size is less than that of the wavelength of the photon. The scattering can occur of atoms or molecules and for molecules the scattering can be inelastic with a change of rotational energy of the molecule.
Compton scattering is inelastic scattering of a photon from a free charged particle. If the charged particle is a bound electron then the energy of the photon must be much greater than the binding energy of the electron.
Side note: Rayleigh scattering is a particular case of Mie scattering. This theory explains in particular the white colour of objects which are made of particles of size greater than the typical wavelength : milk, clouds, chemical powders...
To add to the answer there is Thomson (no "p") scattering which is the elastic scattering of electromagnetic radiation by a free charged particle, as explained by classical electromagnetism.
It is just the low-energy limit of Compton scattering when the particle kinetic energy and photon frequency do not change as a result of the scattering.
This limit is valid as long as the photon energy is much smaller than the mass energy of the particle.