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Condensed Abstract: We present an extensive study of the inception of supernova explosions by following the evolution of the cores of two massive stars (15 Msun and 25 Msun) in two dimensions. Our calculations begin at the onset of core collapse and stop several 100 ms after the bounce, at which time successful explosions of the appropriate magnitude have been obtained. (...) Guided by our numerical results, we have developed a paradigm for the supernova explosion mechanism. We view a supernova as an open cycle thermodynamic engine in which a reservoir of low-entropy matter (the envelope) is thermally coupled and physically connected to a hot bath (the protoneutron star) by a neutrino flux, and by hydrodynamic instabilities. (...) In essence, a Carnot cycle is established in which convection allows out-of-equilibrium heat transfer mediated by neutrinos to drive low entropy matter to higher entropy and therefore extracts mechanical energy from the heat generated by gravitational collapse. We argue that supernova explosions are nearly guaranteed and self-regulated by the high efficiency of the thermodynamic engine. (...) Convection continues to accumulate energy exterior to the neutron star until a successful explosion has occurred. At this time, the envelope is expelled and therefore uncoupled from the heat source (the neutron star) and the energy input ceases. This paradigm does not invoke new or modified physics over previous treatments, but relies on compellingly straightforward thermodynamic arguments. It provides a robust and self-regulated explosion mechanism to power supernovae which is effective under a wide range of physical parameters.