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To measure a galaxy's speed, astronomers spread out its light into a spectrum (rainbow). This reveals the range of wavelengths - from 'short' (blue) to 'long' (red) - emitted by its stars. The spectrum is crossed by dark absorption lines, which correspond to the absorption of light at particular wavelengths by chemical elements in the stars' atmospheres.
If the galaxy is moving, the position of these lines changes. A galaxy approaching us appears to have its wavefronts 'bunched up', and so its lines shift towards the blue, short-wavelength end of the spectrum. Most galaxies are moving away, and so their trailing wavefronts shift the lines to the red. The degree of this shift gives the galaxy's speed.