Redshift
Redshift is the stretching of light to longer, redder wavelengths, which shows that a galaxy or star is moving away from the observer.
Definition
Redshift is the stretching of light from something moving away from us, which shifts the pattern of colors in its light toward the red end of the spectrum; light from an approaching object is blueshifted instead. Astronomers detect it by comparing spectral lines in a galaxy's light with the same lines measured in a lab, and almost all distant galaxies show redshift, which is the evidence that the universe is expanding. It is defined as $z = (\lambda_{\text{observed}} - \lambda_{\text{emitted}})/\lambda_{\text{emitted}}$, and for small speeds it follows the Doppler relation $z \approx v/c$. Cosmological redshift is not motion through space but the stretching of wavelengths by the expansion of space itself, $1 + z = a(t_{\text{now}})/a(t_{\text{emitted}})$, and gravitational redshift is a third kind caused by light climbing out of a gravitational well.
Example
A siren drops in pitch as an ambulance drives away, and light shifts toward red in a similar way when its source recedes. Hydrogen emits a red line at $656\ \text{nm}$ in a lab; in a distant galaxy that line might arrive at $670\ \text{nm}$, showing the galaxy is receding. A galaxy whose hydrogen-alpha line is observed at $721.9\ \text{nm}$ instead of $656.3\ \text{nm}$ has $z = 65.6/656.3 \approx 0.10$, and the most distant confirmed galaxies have $z$ above $10$, meaning their light has been stretched to more than $11$ times its emitted wavelength.
Key Insight
Redshift does not mean a galaxy looks red to your eyes; it means the whole pattern of its spectral lines has moved slightly toward red. The larger the redshift, the farther away the galaxy and the further back in time we are seeing it, so redshift is both a speedometer and a distance ruler. At high redshift, $z$ no longer translates into a simple velocity: a galaxy at $z = 10$ is not moving at ten times the speed of light, but its light was emitted when the universe was $11$ times smaller than it is now.