In a Type Ia supernova, the supernova process happens when the white dwarf in the binary accretes too much mass (anything over about 1.44 times the mass of our sun).  The exact cause of the explosion is still an active area of research, but many think that the extra mass makes the core of the white dwarf heat up, which leads to too much pressure and energy inside the star that it is no longer able to support, and the star violently explodes.
In the core-collapse supernova type, the beginning of the supernova is marked by when the core of the star starts fusing silicon into iron.  Usually, when elements are fused into heavier elements, energy is released, and this energy is what keeps the star from falling in on itself.  Iron, however, is a special element in that it needs to take in energy to be fused into something else.  Once the star starts making iron, the iron starts taking in energy and the star begins to fall in on itself.  The star quickly (~1s) collapses and once the core hits a critical density gravity is overpowered by the nuclear force which becomes repulsive, and the matter is violently expelled outward.