The source of Sun's incessant energy is hydrogen; which is continuously converting to helium through nuclear fusion reaction releasing energy. Why does not all hydrogen convert into helium in one big explosion just like a man-made bomb or cracker explodes?
The reaction rate doesn't increase that quickly with temperature, but pressure does. If you perturb a solar model, making a zone near the core marginally hotter, the increased pressure will rapidly (at roughly the soundspeed divided by a characteristic length) cause it to expand. That lowers the pressure and temperature enough to substantially quench the reaction. So for normal stars during the long early phases of their evolution, the stability of thermonuclear burning is actually very high. There are some cases in stellar eveolution where pressure becomes insensitive to temperature, generally when the density is so high that electrons average kinetic energy is (almost) independent of temperature. In which case, fusion reactions can accelerate dramatically, but in most cases once the temperature is high enough to increase the pressure, the affected parts of the star rapidly expand and quench the reaction. When a sunlike star intitiates Helium fusion such a runaway reaction referred to as a Helium flash ocurrs, but it is easily contained. On the other hand, when a white dwarf accumulates a critical mass of material to intiate carbon burning, the reaction isn't contained and the star is destroyed in a class 1-a supernova. As dmkee points out, this differs from the core collapse supernova of massive stars.
A useful thing to remember is the virial theorem. If the star (or a portion of it) expands because of an increase in temperature, then once the system reaches hydrodynamic equilibrium the temperature must decrease, as the virial theorem determines that kinetic energy is proportional to gravitational binding energy. Thus unless the reaction=>heating=>reaction feedback is faster than the time to reach hydrostatic equilibrium, the star should be stable.
The temperature & pressure required to sustain nuclear fusion reaction are very high and is only present at a small diameter sphere (in comparison to sun's diameter) at its core. Due to massive convection currents in and out of this sphere a fair amount of hydrogen keeps coming to this sphere to sustain the nuclear fusion.