What is the most fundamental property of a star in determining its evolution? luminosity temperature composition mass size?

The most fundamental property of a star that determines its evolution is the MASS of the star.

In fact, while low-mass stars usually end their life becoming red giants and then white or black dwarfs, the high-mass stars can evolve into red supergiants and eventually originate a supernova. After the explosion of the supernova, depending on their mass, they can becomes neutron stars or collapse into black holes.