After the first photons of light are produced, which process is responsible for amplification of the light?

The word "laser" is actually an acronym for light amplification by stimulated emission of radiation. Atoms or molecules of the active medium that have been excited to a higher energy level are stimulated by a passing photon to relax to a lower energy level and emit a photon that is indistinguishable from the passing photon, thereby increasing the number of photons like the incident one.