  If the neutrino luminosity from the proto-neutron star formed during a
massive star core collapse exceeds a critical threshold, a supernova (SN)
results. Using spherical quasi-static evolutionary sequences for hundreds of
progenitors over a range of metallicities, we study how the explosion threshold
maps onto observables, including the fraction of successful explosions, the
neutron star (NS) and black hole (BH) mass functions, the explosion energies
(E_SN) and nickel yields (M_Ni), and their mutual correlations. Successful
explosions are intertwined with failures in a complex pattern that is not
simply related to initial progenitor mass or compactness. We predict that
progenitors with initial masses of 15 +/- 1, 19 +/- 1, and 21-26 M_Sun are most
likely to form BHs, that the BH formation probability is non-zero at
solar-metallicity and increases significantly at low metallicity, and that low
luminosity, low Ni-yield SNe come from progenitors close to success/failure
interfaces. We qualitatively reproduce the observed E_SN-M_Ni correlation, we
predict a correlation between the mean and width of the NS mass and E_SN
distributions, and that the means of the NS and BH mass distributions are
correlated. We show that the observed mean NS mass of ~1.33 M_Sun implies that
the successful explosion fraction is higher than 0.35. Overall, we show that
the neutrino mechanism can in principle explain the observed properties of SNe
and their compact objects. We argue that the rugged landscape of progenitors
and outcomes mandates that SN theory should focus on reproducing the wide
ranging distributions of observed SN properties.
