  We investigate the criteria for successful core-collapse supernova explosions
by the neutrino mechanism. We find that a
critical-luminosity/mass-accretion-rate condition distinguishes non-exploding
from exploding models in hydrodynamic one-dimensional (1D) and two-dimensional
(2D) simulations. We present 95 such simulations that parametrically explore
the dependence on neutrino luminosity, mass accretion rate, resolution, and
dimensionality. While radial oscillations mediate the transition between 1D
accretion (non-exploding) and exploding simulations, the non-radial standing
accretion shock instability characterizes 2D simulations. We find that it is
useful to compare the average dwell time of matter in the gain region with the
corresponding heating timescale, but that tracking the residence time
distribution function of tracer particles better describes the complex flows in
multi-dimensional simulations. Integral quantities such as the net heating
rate, heating efficiency, and mass in the gain region decrease with time in
non-exploding models, but for 2D exploding models, increase before, during, and
after explosion. At the onset of explosion in 2D, the heating efficiency is
$\sim$2% to $\sim$5% and the mass in the gain region is $\sim$0.005 M$_{\sun}$
to $\sim$0.01 M$_{\sun}$. Importantly, we find that the critical luminosity for
explosions in 2D is $\sim$70% of the critical luminosity required in 1D. This
result is not sensitive to resolution or whether the 2D computational domain is
a quadrant or the full 180$^{\circ}$. We suggest that the relaxation of the
explosion condition in going from 1D to 2D (and to, perhaps, 3D) is of a
general character and is not limited by the parametric nature of this study.
