At this point I want to distinguish between stars with masses less than or greater than about 5 solar masses. The actual dividing line is quite uncertain because current stellar models are not sophisticated enough to keep track of all the complicated physics going on.
Anyway, stars smaller than about 5 solar masses (I'll call them low mass stars) never get hot enough to get fusion in their C-O cores at the end of their red giant phase. The collapse of the core therefore continues until a new kind of pressure is able to support the star against gravity. This pressure is called electron degeneracy pressure.
Electron degeneracy pressure is a consequence of an important principle in quantum mechanics called the Pauli exclusion principle. It basically says the following: no two electrons in a material can have exactly the same quantum coordinates. If you try to squeeze a lot of electrons in a small volume, they are forced into a smaller range of space-type quantum coordinates. This makes them be in higher and higher quantum energy levels, so they will tend to move at higher and higher ``velocities.'' This has nothing to do with temperature: even at absolute zero (0 K), the electrons must continue whizzing around, because the Pauli exclusion principle forbids them from coming to rest. This constant motion of electrons produces a pressure which increases with density; thus, it only becomes important in very dense matter.
You can think about electron degeneracy pressure like a crowded freeway. The more crowded a freeway becomes, the harder it is to get on. In a sense, the freeway exerts a pressure on cars trying to merge on.
How dense does the C-O core become? About 
g (1 ton) per cubic centimeter! This hot, superdense core is called a white
dwarf. In the final stages of a low-mass star's life, the outer layers of
the star become unstable and get ejected into space, forming a planetary
nebula. Fusion in the hydrogen and helium shells stops and the white dwarf
becomes exposed, slowly cooling because it is no longer generating energy. Since
degeneracy pressure is unaffected by temperature, the white dwarf does not contract
as it cools; it basically cools off like a charcoal briquet, eventually becoming
a cold black dwarf.
Start:Stars
