  We systematically examine how the presence in a binary affects the final core
structure of a massive star and its consequences for the subsequent supernova
explosion. Interactions with a companion star may change the final rate of
rotation, the size of the helium core, the strength of carbon burning and the
final iron core mass. Stars with initial masses larger than \sim 11\Ms that
experiece core collapse will generally have smaller iron cores at the time of
the explosion if they lost their envelopes due to a previous binary
interaction. Stars below \sim 11\Ms, on the other hand, can end up with larger
helium and metal cores if they have a close companion, since the second
dredge-up phase which reduces the helium core mass dramatically in single stars
does not occur once the hydrogen envelope is lost. We find that the initially
more massive stars in binary systems with masses in the range 8 - 11\Ms are
likely to undergo an electron-capture supernova, while single stars in the same
mass range would end as ONeMg white dwarfs. We suggest that the core collapse
in an electron-capture supernova (and possibly in the case of relatively small
iron cores) leads to a prompt explosion rather than a delayed neutrino-driven
explosion and that this naturally produces neutron stars with low-velocity
kicks. This leads to a dichotomous distribution of neutron star kicks, as
inferred previously, where neutron stars in relatively close binaries attain
low kick velocities. We illustrate the consequences of such a dichotomous kick
scenario using binary population synthesis simulations and discuss its
implications. This scenario has also important consequences for the minimum
initial mass of a massive star that becomes a neutron star. (Abbreviated.)
