A Core Collapse Supernova (Type II Supernova) is a Supernova caused by the collapse of the core of a star. It is identified by Hydrogen Spectral Lines and happens to stars with 8 to around 60 Solar Masses. They are generally observed in HII Regions and spiral arms.
A Fallback Supernova is a supernova (typically core collapse) which leaves some of the expelled material gravitationally bound, to be subsequently accreted. This is theorized to be the cause of some observed Light Curves over the first few weeks.
As of 2017, Core Collapse Supernovae have yet to be convincingly simulated: if sufficiently large and fuel-depleted, a simulated star has collapsed into a Stellar Remnant, but the simultaneous generation of an explosion large enough to qualify as a supernova (on the order of 1 Foe) apparently depends on exotic processes, apparently requiring advances in simulations. Processes include Neutrino interactions, and perhaps General Relativity. Simulation trials support the idea that a supernova may not always happen with a core collapse, depending on convection, Turbulence, spin, and magnetism.