According to the Hertzsprung-Russell diagram, a red giant is a large non-main sequence star of stellar classification K or M; so-named because of the reddish appearance of the cooler "giants". Examples include Aldebaran and Arcturus.
They are believed to be stars of solar mass or higher which have exhausted the supply of hydrogen in their cores and started burning hydrogen in a shell outside the core. Since the source of energy is closer to the surface, the star begins to expand. This makes the star more luminous but, counterintuitively, also reduces the effective temperature. This is because the radius (and hence surface area) of the star increases by a larger amount than the luminosity of the star. As a result, the star becomes larger, but cooler and redder -- hence red giant.
If the star is less than 2.5 solar masses, the addition of helium to the core by shell hydrogen burning will cause a helium flash -- a rapid burst of helium burning in the core, after which the star will commence a brief period of helium burning before beginning another ascent of the red giant branch. Stars more massive than 2.5 solar masses enter the helium burning phase of their lives much more smoothly. The core helium burning phase of a star's life is called the horizontal branch in metal-poor stars, so named because these stars lie on a nearly horizonal line in the Hertzsprung-Russell diagram of many star clusters. Metal-rich helium-burning stars do not lie on a horizonal branch, but instead lie in a clump (the red clump) in the Hertzsprung-Russell diagram.