Doppler shifting - wavelength emitted by something moving away from us is shifted to a lower frequency
Sound of a fire truck siren - pitch of the siren is higher as the fire truck moves towards you, and lower as it moves away from you
Visible wavelengths emitted by objects moving away from us are shifted towards the red part of the visible spectrum
The faster they move away from us, the more they are redshifted. Thus, redshift is a reasonable way to measure the speed of an object (this, by the way, is the principal by which radar guns measure the speed of a car or baseball)
When we observe the redshift of galaxies outside our local group, every galaxy appears to be moving away from us - universe is expanding.