Here's the basics...
Stars are extremely massive. Their gravity is constantly trying to pull all of the mass to the centre of the star. The reason why this doesn't normally happen is because the nuclear fusion going on inside the star provides an outward force that counteracts the force of gravity. The star is constantly in a state of equilibrium as the gravity and fusion pressure are balanced.
Now, when a star starts to run out of hydrogen in it's core, the outward pressure is diminished a bit. Gravity compresses the star a bit, and usually some more hydrogen will burn. Basically, the star will oscillate for a while as it's hydrogen supply sputters. Once the hydrogen is pretty much gone from the core, no fusion happens, and the star begins to collapse rapidly. Pressure in the core will be increased and will have sufficient energy for helium fusion. The helium burning stage will be similar to the hydrogen burning stage (on the inside at least), and this will continue throughout the elements until...
...the star tries to burn iron. The fusion of iron is endothermic, so even though some of it may be fused, all it does is take up energy. The outward force from the core is stopped almost completely, and you've just got a couple umpteen-jillion tons of star collapsing on to an iron core in a free fall. When the star collapses onto it's core (assuming it's massive enough of course), the impact may be intense enough to break down the electron shells and force the electrons into the nucleus. The atom becomes just a core of neutrons, and a neutron star is formed. If there is even more gravitational energy left over, the core may compress even more, to the point where the neutrons are so densely packed that the escape velocity of the remaining "star" is higher than c.
