When a body that is emitting radiation has a non-zero radial velocity relative to an observer, the wavelength of the emission will be shortened or lengthened, depending upon whether the body is moving towards or away from an observer. This change in observed wavelength, or frequency, is known as the Doppler shift.
If the object is moving towards an observer, then the emission will be blueshifted - i.e. the wavelength of the emission will be shortened, moving it towards the blue end of the spectrum. If the object is moving away from an observer, then the emission will be redshifted - i.e. the wavelength of the emission will increase, moving it towards the red end of the spectrum.
In fact, this phenomenon will occur with any type of wave emitted from a moving body - for example, sound waves. This is noticeable for example, when listening to the sound of a siren on a fire engine - as the fire engine approaches, the siren has an increased pitch (as the sound waves have been shortened, resulting in a higher frequency), and as the firetruck recedes, the siren will decrease in pitch (as the sound waves have been lengthened, resulting in a lower frequency).
A Doppler shift is observed in many astronomical objects - particularly in binary or multiple systems where one or more objects are orbiting one another. For example, one method of detecting planets, or faint companion stars, is by searching for a Doppler shift in the spectral line emission from the star as a function of time.
Doppler shift is different to cosmological redshift, which is caused by the expansion of the Universe, and gravitational redshift which is due to a strong gravitational field causing a photon to lose energy before it reaches us.