If however the star's corpse mass is above 1.4 solar masses, it will explode when all fusion ceases. If the corpse mass is exactly 1.4 solar masses, the explosion is called a Type 1A Supernova. Type 1A Supernovae are quite useful, as their intrinsic brightness is well known. As such, when they explode, they can be used as "standard candles" to determine how far away the explosion occurred. A recent survey of Type 1A Supernovae revealed that the expansion rate of the Universe is increasing.
If a star's corpse mass is between 1.4 and 2.8 solar masses, the exploding star becomes a neutron star. The 2.8 solar mass limit is called the Oppenheimer-Volkoff Limit. A neutron star is entirely supported against the force of gravity by so-called neutron degeneracy pressure.
If a star's corpse mass is above the Openheimer-Volkoff Limit, the exploding star is expected to become a black hole. Black holes are exotic objects that are so dense that their escape velocities exceed the speed of light. Since light particles respond to the force of gravity, even light cannot escape the gravitational clutch of a black hole, hence the object's name.
Black holes consist of an extremely (perhaps infinitely) dense matter core, and a sphere around it called the event horizon. Any matter or light particles that cross the event horizon going towards the black hole cannot escape the black hole: the event horizon distance, called the Schwarzchild Radius, is that distance from the black hole's center where the escape velocity becomes equal to the speed of light.
There is some speculation, but no proof, that there are other degeneracy pressures, specifically quark and preon degeneracy pressures, that kick in at pressures exceeding that which neutron degeneracy pressure can withstand. Perhaps these possible degeneracy pressures prevent black holes from achieving infinite density.
If a star's corpse mass is less than the Chandrasekhar Limit of 1.4 solar masses (a solar mass is the current mass of the Sun), the star dies without exploding: it becomes a white dwarf. A white dwarf is a star remnant entirely supported against the crushing force of gravity by so-called electron degeneracy pressure. It is quite hot, and stays that way for a very long time. It's also quite dim, shrunk to the size of the Earth.
In a star's dying phase, it swells up due to the pressure action of shell fusion. The farther out the concentric shell being fused, the more the star is swollen in size. In this swollen state, the star often "shucks off" some of its outer atmosphere, reducing its overall mass. The key measurement here is the mass a star has at the moment it dies: this is the star's corpse mass.
Stars commence their "lives" when core fusion starts, and they start dying when shell fusion commences. Stars officially die when all fusion processes within them cease.