The term “sooty star” or “carbon star” almost makes them sound like big chunks of coal.  You know that wouldn’t work.  In the interest of brevity, quick definition of a carbon star is an evolved giant (or sometimes a dwarf) with circumstellar “clouds” of carbon dust.  It’s seeing the star through this atmosphere of carbon that gives it its red appearance; because carbon absorbs “blue” light.  Classical carbon stars are massive; non-classical less so.  Most are long-period variables.

Image:  Greg Parker, New Forest Observatory

This is an image of La Superba – Y Canum Venaticorum (Y CVn), one of the reddest stars in the sky, and among the brightest of the red giants.  It’s a semi-regular variable carbon star.

Image:  H. Olafsson, Stockholm Observatory (et al)

This is TT Cygni.  The ring you see around the star is a shell of gas expanding outward.

Classical carbon stars are single giants at the end of their lifespan.  Non-classical carbon stars are those in a binary system where one is a white dwarf and the other a red giant.  The giant accreted carbon from its companion, now the dwarf, to become the carbon star.

Carbon stars are somewhat rare, and they quickly evolve out of this phase because they lose tremendous amounts of mass.  Their carbon “shell” becomes part of the interstellar dust; providing the raw materials for the creation of new generations of stars and planets.