A white dwarf is an extremely compact remnant left behind by a star of around 8 - 11 solar masses at the end of its life. White dwarfs generally have masses between 0.5 and 1.4 solar masses with a radius (and therefore volume) similar to that of the Earth. Their position at the bottom of the CMD denotes their faint luminosity (perhaps 0.1 - 1 % that of our Sun). Our nearest white dwarf is the invisible (at least in the optical) companion to Sirius, known as Sirius B.
Despite having no nuclear reactions in their cores, they start life with very high surface temperatures, perhaps as high as 100,000 K. This is because the collapse of outer layers of the progenitor star (the normal star before it became a white dwarf) down onto the stellar core converts a huge amount of gravitational potential energy into heat. With no further energy being generated, it might be expected that the temperature would drop rapidly, but this is not the case. White dwarfs do eventually cool, but their small surface gives them a low luminosity and hence they radiate their heat away very slowly taking billions of years to cool down to temperatures near 10,000 K.
The material within a white dwarf becomes degenerate (degeneracy is a result of the Pauli exclusion principle; no two electrons can occupy identical quantum states, despite being under the pressure of a collapsed star) and is at densities of about 109 kg m-3. This is equal to 1 million tonnes per cubic metre and is about 200,000 times as dense as the Earth!
One strange property of white dwarfs, which is determined by their degenerate nature, is their mass-radius relationship (see Figure 1). The normal relationships between temperature, pressure and density do not hold for degenerate matter. As the mass of a white dwarf increases, its radius decreases. As you can see, there is a maximum mass beyond which the white dwarf becomes unstable - in theory, this is the point at which their radius decreases to zero. This limit (known as the Chandrasekhar Limit) was calculated by Subrahmanyan Chandrasekhar in the 1930s to be approximately 1.44 solar masses.