Definitions for "Red giant"
A relatively old star whose diameter has swollen enormously. It's temperature has also cooled appreciably, it's contracting hydrogen core has turned to helium and eventually to carbon. Our Sun will become a red giant star in about 5 billion years.
A large, red sunlike star in its latter stages of life.
A stage in the life-cycle of low-mass and medium-mass stars. After a star's energy runs out, it cools down and swells up into a red giant.
The shift of spectral lines to longer wavelengths either due to the motion of the source away from the observer or very strong gravity.