Since we know that the Universe is expanding, we expect gravitationally unconnected galaxies, and galaxy clusters, to show a "red shift" in the absorbed lines of their spectra, and they do. This information can be used to tell how far away a galaxy, or galaxy cluster, is.
In astronomy, if we are looking at an absorption spectrum of a moving light source, say a star, the missing frequencies, the absorbed frequencies won't be in the right place. They will have shifted, changed colors. We can tell what they should have been, if there was no relative movement of the light source, because the lines have a relationship to each other, depending on the chemical composition of the intervening surface gas of the star. The amount of frequency shift for any absorbed frequency tells us the speed and the direction the star is moving at, relative to us here on Earth.
Note that the speed of light doesn't change: that's not allowed. Only the frequencies of the light shift.
The amount of doppler induced frequency shifting of a light wave depends on how fast the light source is moving, relative to the observer. The faster the speed of the light source towards or away from the observer, the greater the frequency shift.
The Doppler Effect applies to all wave phenomena, including light. When a light source is moving towards an observer, all of its frequencies shift up. Put another way, all visible colors shift towards the blue end of the visible portion of the electromagnetic spectrum. The opposite is true as well: when a light source is moving away from an observer, all its frequencies shift towards the red end of the visible portion of the electromagnetic spectrum.
The Doppler Effect is something that everyone has experienced. When an ambulance, or in the case of the image to the left, a train mores towards an observer, its horn sounds at a higher pitch than normal. The opposite is true as well: when the same train moves away from an observer, the horn's pitch is lower than normal.