P.Bo's description was good, but I wanted to add a few things.
When you compress a gas, that gas heats up. When it expands, it cools down. This has been known for a long time and is very widely used in all kinds of things such as air conditioners. In space it is no different. When a cloud of gas undergoes collapse, it must heat up. This increases the pressure of the gas cloud on itself because the atoms and molecules are moving faster when they heat up and exerting said pressure. Now, for the cloud to collapse, it must get rid of that heat. How does it do this? It radiates it away in the form of light, IR, or other electromagnetic radiation. This reduces the temperature, and thus the outward pressure of the gas, allowing it to collapse.
How does this relate to a star? Well, a star is basically a giant ball of compressed gases. The core is compressed to extreme temperatures and pressures because the star's outer layers, which have more mass than thousands of Earths, are pressing down upon it. So what happens? The core, thanks to the temperature, generates an extreme amount of high energy EM radiation. (In addition to the normal gamma rays and such from the fusion events) This EM radiation carries energy out from the core and into the rest of the star, eventually escaping through the photosphere of the star.
Now, remember what I said up above. When that gas cloud loses energy through EM radiation, the pressure inside is reduced, right? So what happens to the core? If we DIDN'T have nuclear fusion the core would constantly shrink in size as energy is radiated out from the star. But the fusion replaces this energy lost at an equal rate, which means the core doesn't shrink. Eventually, when the fuel supply for fusion runs out, the core can no longer maintain its size and starts to collapse.
Uh oh, its a gas and its collapsing. That means that the temperature is increasing again! If the star is large enough the collapse can increase the temperature so high that it can use the next element up from Hydrogen, which is Helium, as a fuel source for yet another round of nuclear fusion. At least until it runs out too. Very massive stars go through many cycles of this and build up heavier and heavier elements in their core. Once fusion in each round starts, the core stops collapsing as the energy lost through radiation is replaced by fusion, only to start collapsing again once the fuel runs out and fusion ceases.
Once we reach nickel as products of fusion, it's the death sentence for the star. Attempting to fuse nickel with itself does NOT give you energy. It TAKES energy. So once you have nothing but nickel inside a star's core, there is nothing the star can do to replace the energy lost. So again, the star runs out of fuel once nickel is built up and starts to collapse, heating the core up again.
Now, at a certain point, the core simply cannot collapse any further. The atoms cannot get any closer together because all the electrons repel each other and cannot occupy the same spot. But as the core of nickel builds up, the pressure increases to the point where even this 'degeneracy pressure' can't withstand the crushing weight of the star upon itself. The electrons combine with protons and form neutrons, a process with TAKES energy away from the core. This only causes further collapse and a chain reaction starts that results in the creation of a core collapse supernova.
http://en.wikipedia.org/wiki/Type_II_supernova
I hope that helps.
