What is a dying star called?

A dying star is a called a "red giant." This is a star whose diameter is many times greater than that of the sun (despite the fact that it is cooler). The brightness of a red giant comes from its shell, which is still burning hydrogen.

Regular or "main sequence" stars fuse hydrogen into helium. Once the fusion ends, the star begins to contract. This causes temperatures in the star to rise until the helium in the core fuses into carbon. Depending on the size of the star, it either explodes into a supernova or bloats into a red giant. Eventually, many red giants degrade into white dwarves found within planetary nebula.