Gravity always tries to collapse the mass of a star toward its center. What mechanism can oppose this gravitational collapse for a star? During what stages of a star's life would there be a "balance" between them?

The gravity collapsing the core, increases pressure causing the fusion of hydrogen to helium. The newly formed helium creates outward pressure. The pressure balances the gravitational inward pull. This is the hydrostatic equilibrium and it occurs throughout the main sequence phase.