What determines the fate of a star?

a star: After millions to billions of years, depending on the initial mass of the star, the continuous fusion of hydrogen into helium will cause a build-up of helium in the core. Larger and hotter stars produce helium more rapidly than cooler and less massive ones. The accumulation of helium, which is denser than hydrogen, in the core causes gravitational self-compression and a gradual increase in the rate of fusion. Higher temperatures must be attained to resist this increase in gravitational compression and to maintain a steady state.

Eventually, the core exhausts its supply of hydrogen, and without the outward pressure generated by the fusion of hydrogen to counteract the force of gravity, it contracts until either electron degeneracy becomes sufficient to oppose gravity, or the core becomes hot enough (around 100 megakelvins) for helium fusion to begin. Which of these happens first depends upon the star's mass.