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In physics (especially astrophysics), redshift happens when light seen coming from an object that is moving away is proportionally increased in wavelength, or shifted to the red end of the spectrum. More generally, when an observer detects electromagnetic radiation outside the visible spectrum, "redder" amounts to a technical shorthand for "increase in electromagnetic wavelength" — which also implies lower frequency and photon energy in accord with, respectively, the wave and quantum theories of light.
Redshifts are attributable to the Doppler effect, familiar in the changes in the apparent pitches of sirens and frequency of the sound waves emitted by speeding vehicles; an observed redshift due to the Doppler effect occurs whenever a light source moves away from an observer. Cosmological redshift is seen due to the expansion of the universe, and sufficiently distant light sources (generally more than a few million light years away) show redshift corresponding to the rate of increase of their distance from Earth.
HOWEVER - the speed of light HAS to remain constant. You ARE seeing a dopplar effect, but you are seeing a shift in wavelength, not speed!