What happens to that material depends on how much mass there is. If you end up with a stellar remnant that's about the mass of our Sun, they call it a white dwarf star. If you go bigger, there's enough gravity to create a neutron star. Go bigger than that, and there's so much gravity, it all collapses into a black hole. (Contrary to popular belief, a black hole isn't actually a hole. It's the most filled-up hole in existence, so dense is the matter within it.)
So, how much matter will remain as a white dwarf, and how much will become a neutron star? The difference is called the Chandrasekhar Limit. (Named after an astrophysicist called Subrahmanyan Chandrasekhar, a Nobel prize winner who got routinely snubbed early on in his career despite his brilliance.)
Currently, we believe the Chandrasekhar Limit for a white dwarf is about 1.4 solar masses (nearly one-and-a-half times the size of our Sun), or about 2.765 × 1030 kg. That's about as big as they can get and still be called a white dwarf.
If the stellar remnant is under this mass, it'll be a white dwarf star. (Okay, white dwarves aren't really "stars", in that they don't glow under their own power. What you see is the heat radiating off it as it cools down into a cold cinder. There's an awful lot of heat, so this will take quite some time.) Electron degeneracy pressure is what keeps it from collapsing entirely on itself.
If the stellar remnant is over this mass, it collapses down into a neutron star, maintained at a radius of about 10 kilometers by neutron degeneracy pressure. But if you get enough mass to overcome neutron degeneracy pressure, the stellar remnant collapses so much it become a singularity, or a black hole. There, the gravity is so strong it will bend light.
So, a star dies and you end up with a white dwarf. The real fun with a Chandrasekhar Limit comes if the white dwarf is part of a binary system. If it's got a mate that's real close, and they
A white star can only get as big as 1.4 solar masses. What happens if the white dwarf sucks off extra matter from its companion or two white dwarves merge?
It goes BOOM! When the matter of a white dwarf exceeds the Chandrasekhar Limit, you get a type Ia supernova. This is where the star gets enough mass to ignite carbon fusion, and things go ker-splodey. In this case, the white dwarf is a goner.
Go hardcore: learn more about how the Chandrasekhar Limit affects White Dwarf stars.
What do you think a white dwarf will look like once it eventually cools?
Her Grace has not yet had an opportunity for the direct observation of a white dwarf, but she'd like to get a gander at Sirius B.