All massive stars end their lives with core collapse and many as supernova explosions. Despite observations of thousands of supernovae, detailed numerical calculations and theoretical efforts, the mechanism of explosion is poorly understood and perhaps even unknown. By parameterizing the systematic uncertainty in the explosion mechanism and by using spherical quasi-static evolutionary sequences for many hundreds of progenitors over a wide range of metallicities, we study how the explosion threshold maps onto observables - fraction of successful explosions, remnant neutron star and black hole mass functions, explosion energies, nickel yields - and their mutual correlations. Successful explosions are intertwined with failures in a complex but well-defined pattern that is not well described by the progenitor initial mass and is tied to the pre-collapse structure of the progenitor star. Other supernova properties show a similar pattern. Finally, to facilitate better comparison of the theory and the data, we present a new method to extract parameters from supernova light curves and expansion velocities in a statistically correct way.