Red Giant
A red giant is a large, cool, luminous star late in its life that has used up the hydrogen in its core and expanded to many times its original size.
Definition
A red giant is an old star that has swelled to many times its original size, with outer layers that have spread out and cooled so that it glows reddish. It forms when a star exhausts the hydrogen in its core and moves off the main sequence: the core shrinks and heats while hydrogen fuses in a shell around it, and the extra energy pushes the outer layers out to tens or hundreds of times the original radius, cooling the surface to around $3{,}000$ to $4{,}000^\circ\text{C}$. Red giants come from stars of roughly $0.3$ to $8$ solar masses, with hydrogen shell burning around an inert, contracting helium core. Helium ignition, which occurs as a helium flash in stars below about $2$ solar masses because the core is electron-degenerate, moves the star to the horizontal branch, and a later asymptotic giant branch phase precedes ejection of the envelope as a planetary nebula.
Example
Arcturus, the bright orange star of the spring sky, is a red giant about $25$ times wider than the Sun. In about $5$ billion years the Sun will expand to roughly $200$ times its current radius, reaching past Earth's present orbit and shining thousands of times brighter, swallowing Mercury and Venus along the way. Aldebaran, the eye of Taurus, is a red giant of about $1.2$ solar masses with a radius near $44$ solar radii, a luminosity around $400$ times the Sun's, and a surface temperature around $3{,}600^\circ\text{C}$.
Key Insight
A red giant's surface is cooler than the Sun's, yet it is far brighter, because it has so much more glowing surface area. It is red because it is cool but giant because its core has become hotter, two facts that seem opposite but both come from running out of core hydrogen. The red giant phase is also where a star loses most of its mass: strong winds and the final planetary nebula return most of a Sun-like star's envelope to space, enriched with the carbon and nitrogen made during its giant years.