When a light source is moving away or towards us its frequency appears to change.
When we look at light from the sun some freqencies of light are not seen because they are absorbed by hydrogen and helium in the sun. This appears as black lines in the sun's spectrum.
When we look at a galaxy moving away from us the black lines have been shifted to lower frequencies. This is called the red-shift. If the galaxy was moving towards us it would be the blue-shift.
Almost all glaxies show red-shift. The futher away a glaxy is the more red-shifted it is. This means that the more distant the galaxy the faster it is moving away from us.
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