The effect in which light shifts toward red as its source moves away from an observer is the Doppler effect.
For light, a receding source produces a Doppler redshift: the observed wavelength becomes longer, while the observed frequency and photon energy decrease. An approaching source produces the opposite pattern, called a blueshift. The effect depends on relative motion along the line of sight and is too small to notice by eye in most everyday situations.
The phenomenon is named after Austrian mathematician and physicist Christian Doppler, who explained it in 1842. Astronomers use shifts in familiar spectral lines to measure whether stars and galaxies are approaching or receding, and to estimate their radial velocities. The method can also reveal binary stars, stellar rotation, and some exoplanets.
A useful distinction is that not every redshift is a Doppler shift. Gravitational redshift occurs when light escapes a gravitational field, while cosmological redshift results from the expansion of space. In the wording of this question, the source's motion specifically identifies the Doppler effect.