Supernovae are explosions that are produced when a star can no longer hold itself up against gravity. There are actually two major ways in which stars can produce supernova explosions.
The more well-known way by which a supernova is triggered is for a star to explode after it exhausts all of its thermonuclear fuel. Stars usually remain in a state of balance, in which gravity pulls a star together, whereas the heat generated by fusion in the stellar core resists gravity's attempt to cause the star to collapse.\footnote{To visualize this, you can think of a balloon filled with air. Although mathematically the analogy is poor, the tension of the rubber wants to make the balloon shrink, whereas the pressure from the air holds it up. Where then does heat come in? Heat, or more specifically temperature, and pressure are closely related. If you heat the balloon, the pressure inside will increase, causing it to expand. Likewise, if you cool the balloon, the pressure will decrease, and the tension of the rubber will cause it to shrink. You can try this at home --- inflate a balloon, put it in the freezer, and pull it out in an hour or so. The balloon will have deflated noticeably.} However, the fuel in the star's core --- hydrogen, helium, carbon, and some heavier elements --- is a finite resource. By burning these elements, the star steadily produces heavier elements --- neon, magnesium, and other elements up to iron. When the core of the star is composed of iron, the fuel is effectively gone. The pressure that was resisting the force of gravity disappears, and the star collapses.
We know that this produces a supernova, because we have now seen a few stars that disappeared after a supernova explosion. We also have a lot of indirect evidence that collapsing stars produce supernova, because we have found the elements produced in the core of a star in the remnants of supernova explosions. What we don't really know is why a collapsing star ends up sending so much matter outward in an explosion. It is thought that the explosion is triggered when something halts the collapse of the star.
In fact, we do know that sometimes the collapses of dying stars are halted, because cinders of the former stellar cores have been found at the centers of the remnants of supernova explosions. These cinders are referred to as "neutron stars." Neutron stars are not held up by heat like ordinary stars. Rather, the resistance against gravity is provided by the refusal of neutrons to occupy the same space (I ignore protons, because in the interior of the neutron star, protons merge with electrons to form neutrons). In technical terms, they are held up by the Pauli exclusion principle that underlies quantum mechanics, and by the strong nuclear forces (quantum chromo-dynamics) that govern the structure of atoms. In more visceral terms, they are enormous atoms, compressing a mass equal to that of the sun into a sphere with a radius of about 5 miles. In any case, many astrophysicists suspect that the formation of a neutron star turns the tide on the collapsing star, and causes it to produce an explosion.