White Dwarfs are the remnants of smaller stars, roughly < 3 Solar Masses, left over after their "death", ie the outer layers of the red giant cast off, leaving behind a highly dense white hot core of matter, the exact composition of which changes depending on the original star.
A Black dwarf is merely a white dwarf which has cooled down enough that it emits neglible light, but seeing as this takes so long, there are none in the Universe.
A neutron star is what happens when the star is so very massive it becomes a supernova, which compresses the core of the star so much, the protons and neutrons get pressed very close together. Normally they are kept apart by the Pauli Exclusion principle (this effect in stars is known as degeneracy pressure), but the immense pressures force them so close together, they essentially join and form neutrons, which are not as strongly effected by the principle at those pressure levels, so form a more stable structure.
The Chandrasekhar's limit is essentially the limit on the mass of white dwarfs. If the mass passes a certain limit, the Pauli Exclusion Principle can no longer support the star against the pressure caused by gravity and it becomes a neutron star. You get a similar limit for neutron stars, as if they are too massive you get a black hole.