What happens to light waves from a star as the star moves away from earth?

The astronomer Edwin Powell Hubble observed several celestial bodies, and when obtaining the spectra of distant galaxies he observed the spectral lines were displaced towards the red (red shift), whereas the nearby galaxies showed a spectrum displaced to the blue.  

From there, Hubble deduced that the farther the galaxy is, the more redshifted it is in its spectrum. The same happens with the stars and this phenomenom is known as the Doppler effect.

This phenomenon refers to the change in a wave perceived frequency (or wavelength=color) when the emitter of the waves, and the receiver (or observer in the case of light) move relative to each other.  For example, as a star moves away from the Earth, its espectrum turns towards the red.