What is the most important property of a star that determines its evolution?

The most important property of a star that determines its evolution is its mass.

In fact:
- Low-mass stars becomes red giants, which can collapse (again, depending on the mass) into white dwarfs or into type I-A supernovae
- High-mass stars becomes instead red supergiants, and finally they collapse into type 2-A supernovae. Depending on the mass, the star finally collapses into a neutron star (for stars with medium masses) or into a black hole (for very massive stars).