In the case of a core collapse supernova, it is the core of the star which collapses to form a neutron star, while the outer layers are blasted into space. The scenario goes something like this ...

Nuclear fusion inside a star cannot form any nuclei heavier than iron. So a massive star will build up a central core of mostly iron, "ash" from the nuclear reactions in the rest of the star. The core will have a very high temperature (in excess of 1,000,000,000 Kelvins). Once that core exceeds about 1.4 solar masses (the Chandrasekhar Limit), and temperature about 3,000,000,000 Kelvins, it will collapse from something roughly the size of Earth down to something about the size of a city (20 km or so across). The collapse is catastrophic, and takes about 0.1 seconds to complete. From the point of view of the outer layers, the core has simply & suddenly vanished, and the outer layers respond by falling very fast under the enormous weight of the entire remaining star. The collapsing core, meanwhile, is falling so fast itself that it falls too far and rebounds. As the core rebounds outward, the fast falling stellar stuff rams into it at a significant fraction of the speed of light and the result is a powerful shock wave that propagates outward and eventually blows away most of the star. Material that falls through the shack wave lands on the core, and if it continues to grow and adds enough mass, the neutron star will collapse in turn to a black hole, which does not make an explosion.

That's the way the physics is supposed to work in theory. But the devil is in the details as they say, and so far as I recall, nobody has yet managed to build a computer model of a core collapse supernova that actually explodes. In the computer models the shock wave invariably stalls as it tries to pass through the dense & massive outer layers, so there is no explosion. Obviously we have a few things yet to learn about such things.

You can find all the physical & mathematical detail you could ever hope for in the 172 page review paper Explosion mechanism, neutrino burst and gravitational wave in core-collapse supernovae; Kotake, Sato & Takahashi, Reports on Progress in Physics 69(4): 971-1143, April 2006, plus citations thereto and references therein. See also Theory of core-collapse supernovae; Janka, et al., Physics Reports 442(1-6): 38-74, April 2007.

There is another class of supernova, the type-Ia deflagration supernova. While core collapse supernovae start with a normal star, a type-Ia starts with a white dwarf accreting material from a companion star. A type-Ia supernova (also known as thermonuclear supernovae) leaves no compact core behind, the entire white dwarf is totally destroyed. See, i.e., Roepke, 2008; Iapachino, et al., 2006.

The first generation of stars were made of pure hydrogen & helium and are expected to be very massive compared to stars we see today (Bromm & Larson, 2004). They will generate heavier elements inside via the usual nucleosynthesis, and can explode in the same way. Stars in excess of about 100 solar masses experience supernovae through another mechanism, the pair-production supernova (i.e., Bromm, Yoshida & Hernquist, 2003; Scannapieco, et al., 2005; Scannapieco, 2009).

As you might guess from the accounts above, the mass that winds up trapped in black holes & neutron stars is a miniscule fraction of the total mass involved. The vast majority of the mass is blown away in the supernova explosions. The speed of the outgoing blast is far beyond the limit required to escape the massive core. Besides, as already pointed out by Cougar, massive stars can lose more than half their mass just from the stellar wind pushed out by radiation pressure, although that effect is much weaker for the first generation of stars which have low metal abundances, as the winds are driven by spectral absorption by the metals.

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The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell