We present two-dimensional hydrodynamic simulations of stellar core collapse and develop the framework for a detailed analysis of the energetic aspects of neutrino-powered supernova explosions. Our results confirm that the neutrino-heating mechanism remains a viable explanation of the explosion of a wider mass range of supernova progenitors with iron cores, but the explosion sets in later and develops differently than thought so far. The calculations were performed with an energy-dependent treatment of the neutrino transport based on the "ray-by-ray plus" approximation, in which the neutrino number, energy, and momentum equations are closed with a variable Eddington factor obtained by iteratively solving a model Boltzmann equation. We focus here on the evolution of a 15 M _☉ progenitor and provide evidence that shock revival and an explosion are initiated at about 600 ms after core bounce, powered by neutrino energy deposition. This is significantly later than previously found for an 11.2 M _☉ star, for which we also present a continuation of the explosion model published by Buras et al. The onset of the blast is fostered in both cases by the standing accretion-shock instability. This instability exhibits highest growth rates for the dipole and quadrupole modes, which lead to large-amplitude bipolar shock oscillations and push the shock to larger radii, thus increasing the time accreted matter is exposed to neutrino heating in the gain layer. As a consequence, also convective overturn behind the shock is strengthened, which otherwise is suppressed or damped because of the small shock stagnation radius. When the explosion sets in, the shock reveals a pronounced global deformation with a dominant dipolar component. In both the 11.2 M _☉ and 15 M _☉ explosions long-lasting equatorial downflows supply the gain layer with fresh gas, of which a sizable fraction is heated by neutrinos and leads to the build-up of the explosion energy of the ejecta over possibly hundreds of milliseconds. A "soft" nuclear equation of state that causes a rapid contraction, and a smaller radius of the forming neutron star and thus a fast release of gravitational binding energy, seems to be more favorable for the development of an explosion. Rotation has the opposite effect because in the long run it leads to a more extended and cooler neutron star and thus lower neutrino luminosities and mean energies and overall less neutrino heating. Neutron star g-mode oscillations, although we see their presence, and the acoustic mechanism play no important role in our simulations. While numerical tests show that our code is also well able to follow large-amplitude core g-modes if they are instigated; the amplitude of such oscillations remains small in our supernova runs and the acoustic energy flux injected by the ringing neutron star and by the deceleration of supersonic downflows near the neutron star surface is small compared to the neutrino energy deposition.