The end state of a low mass star is a white dwarf. This is essentially the "ash" core left after nuclear burning has gone as far as it can. In a star less than 8 solar masses, the core will be carbon (with nitrogen and oxygen) left from the triple-alpha reaction and some CNO cycle. The core does not collapse enough to ignite these, because it becomes "degenerate". This means that the free electrons are packed close enough that a quantum principle (which says that no two can occupy the same energy state if they can feel each other) comes into play. Many of the electrons are forced to occupy high energy states, which makes them move very fast and supply enough pressure to hold up the core.
This form of pressure support has 2 strange characteristics: 1) adding heat does not increase the pressure (the electrons are already moving much faster than normal) and 2) adding mass causes the object to shrink in size (giving it higher density and more pressure support).  By this time the core is about the size of the Earth, and typically contains 0.5 to 1 solar mass. For stars which start between 0.5 and 8 solar masses it is made of carbon and oxygen. For stars starting between 8-12 solar masses the core is made of elements between oxygen and iron (but the white dwarf has similar mass as before). The rest of the star has been ejected as a planetary nebula. The density of the core is 10^6-7 gm/cc. Chandrasekhar found that the limiting mass of a white dwarf is about 1.4 solar masses: beyond that electron degeneracy simply cannot resist gravity (and the object has shrunk too far). White dwarfs shine only by radiating away their internal heat, which makes them increasingly faint (here's a picture of some in a globular cluster).

If the white dwarf is in a binary system, its companion (which must have started with less mass) can later evolve to the red giant stage. If the stars are close enough, the surface of the giant can reach the point where gravity is as strong from the white dwarf as from its own center. Material can begin to spill over onto the white dwarf (though generally it will first have to form an "accretion disk" around the white dwarf to get rid of orbital angular momentum). As fresh hydrogen accumulates on the white dwarf, the conditions for fusion can once again occur. A thermonuclear detonation can consume the surface hydrogen suddenly, causing the star to brighten by thousands of times. There is nothing confining the reaction, so it will rapidly blow itself off the star (where we can follow the products in an expanding shell for a while). This event is called a "nova". Since the underlying white dwarf is not damaged, the process can repeat many times.

On the other hand, if the white dwarf binary is near the Chandrasekhar limit, and the red giant dumps material too fast, the white dwarf may be pushed over the limit. It will begin to collapse, and its main body can all begin to fuse again. Since there is nothing holding it in, this will cause the entire white dwarf to explode spectacularly, utterly destroying it. Such explosions (called Type I supernovae) can be as bright as a whole galaxy when they occur. A lot of iron and nickel is produced, much of it radioactive, which keeps the expanding fireball glowing for months. This is the source of the iron/nickel in the Earth's core, and in your blood. We can see these explosions across the Universe, and they seem to have the same energy all the time (because of the limiting mass). We will see that this makes them important for cosmology.

In stars with mass greater than 12 solar masses, the core will eventually burn up to iron, and will come to contain 1.4 solar masses or more. In this case a white dwarf will not form, and the core collapses suddenly (when fusion stops). Dropping in at nearly the speed of light, the electrons now find it more energetically favorable to merge with the protons - making neutrons and emitting neutrinos. The result is a neutron star, about 10 km in radius and having the density of an atomic nucleus (10¹³ gm/cc), supported by neutron degeneracy. Most neutron stars measured have just over 1.4 solar masses. The magnetic fields and the angular momentum of the star that collapsed are greatly concentrated in this process, leaving a rapidly spinning object with intense fields. It can emit beams of radio waves, which if they sweep over the Earth cause us to see very regular pulses (hence the name "pulsar").

The collapse of the core leaves the rest of the star with no pressure support at its center. It begins to flow rapidly towards the center, attaining a good fraction of the speed of light. Then it suddenly encounters the ultradense neutron star, and "bounces", sending a shock wave speeding outward through the envelope. At the same time, the intense flood of neutrinos (which carry 99.9% of the energy of the explosion) is also heading outward. Even though neutrinos barely interact with matter, there are enough of them and enough matter in the envelope (many solar masses) that they provide an outward push too. The combination is enough to blast the entire envelope of the star away, at something like 10,000 km/s. This too is a spectacular explosion almost as bright as a galaxy, called a Type II supernova.

There is so much free energy and so many free neutrons available during the explosion that all the elements heavier than iron can be sythesized in the expanding envelope (though in very small quantities). This is the only source for them in the Universe. The expanding envelope can be visible for tens of thousands of years - it is called a supernova remnant. The pulsar at the center can also supply energetic photon and particle beams to keep the remnant glowing. Eventually the remnant will disperse, seeding the interstellar medium with heavy elements and filling it with very hot hydrogen and helium.

Here's a page with some related animations.

Sometimes the core will retain too much mass to even be supported by neutron degeneracy. This limiting mass is analogous to the Chandrasekhar limit, but is about twice as much. In that case, the collapsing core does not halt at 10 km radius, but continues to shrink. As it passes below the Schwarschild radius (about 3 km), the curvature of spacetime due to gravity causes it to close on itself. Put in a more familiar sounding way, the escape velocity at the surface of the object finally exceeds the velocity of light. Either way, this radius becomes an "event horizon", ie. a horizon over which no further events can be observed. The stellar core has collapsed to become a black hole. In some cases this may occur during the initial collapse, in which case the rest of the star will not "bounce" but will follow the core into the black hole (and there will be no explosion). In other cases the envelope may be partially exploded away, but enough comes raining down on the neutron star after a few hours to push it over the brink.

The black hole has many strange properties. All information about what it has swallowed is lost, except for Mass, Angular Momentum, and Charge. The event horizon is spherical if the hole is not rotating, and surrounded by an egg-shaped "ergosphere" if it is rotating. Space itself is dragged around by the rotation; you would spin around the black hole without feeling any effects of spin. Time (as seen by an external observer) slows down near the hole, and time stops at the event horizon. Light from near the hole is very redshifted as it has to climb out of the gravitational potential well (this is true for white dwarfs and neutron stars too, but extreme for black holes). Images that you can see become increasingly distorted by the curvature of space-time. Near the hole tidal forces become extreme: any matter will be stretched in a radial direction and compressed perpendicular to that, until transformed into a thin stream of atoms. You can orbit the hole without difficulty; the gravity a few thousand miles out is relatively normal (although orbital speeds are very high). But if you don't have any angular momentum you will fall in - moving near the speed of light as you approach the event horizon. There is nothing physical at the horizon, you've merely past the point of no return. Microseconds later you would join all previous matter at the central singularity - a single point of infinite density (actually something else may happen, but current physics doesn't know what). It doesn't matter anyway, no information on what goes on inside the event horizon can ever be known on the outside.

We can observe the presence of neutron stars and black holes by finding them in binary systems. The companion star can be seen, and the mass of the compact object inferred by its orbit. If mass is flowing onto it, it will form an accretion disk. Near the compact object orbital speeds are very high, and the material heats up to millions of degrees, emitting X-rays. Thus, an X-ray binary system in which the unseen companion is between 1.4 and 3 solar masses must have a neutron star, and if the compact object is more massive it must be a black hole. Other behavior of the light from the inner disk also supports this interpretation. In a mechanism perhaps related to the protostellar jets, a relativistic beam of particles is sometimes emitted out the rotational poles of the disk, the extreme energy of the particles also shows that a very compact object is their source.