After a helium-burning red giant runs out of helium fuel in its core, the star's core starts to collapse and heat up. This causes the outer layers of the star to expand and cool, similar to the process that occurred after the star ran out of hydrogen fuel and left the main sequence. As the star swells larger and larger, it eventually becomes a red supergiant.
If the red supergiant is not massive enough, the carbon and oxygen in the core will never get hot enough to fuse into heavier elements. Such will be the case with our own Sun one day.
Eventually the outer layers of the star will be shed, creating a planetary nebula, with only a white dwarf left