What is the climax of a dying star?

This depends on the size of star that's dying. A star the mass of our sun pretty much goes with a whimper. At the very last moments of the star's life, it'll fall in on itself and become a white dwarf, which basically is a glowing ember made of solid iron or carbon. Larger stars have more violent deaths resulting in supernovae and the remnants become either a neutron star or a black hole.