There are 2 main kinds of supernova. The kind you are referring to is a type 2 supernova. There are also type 1a, type 1b, and type 1c supernova. Type 1a supernova are not caused by the implosion of huge stars, but rather, a white dwarf accumulating ever greater amounts of mass from a companion star until the Chandrasekhar limit for mass is reached. Then we get the type 1a supernova, which leaves little core remnant and is more or less constant sized explosion with standard luminosity. This is the reason type 1a supernova are used as standard candles to gauge the distance of objects in the universe.
As David Hammen's answer has pointed out, type 2 supernova endings occur only for stars that are more than 8x our sun's mass. Neutron stars and black holes are remnants left post-supernova, after the outer layers and majority of star has has blown off. The size of the remnant is predictable to an extent pre-supernova and depends on the mass of the star. However this is only an approximation and there are always exceptions as the physics is complex and depends on too many variables. Type 1 supernovas leave little of no remnant.
Cause of Type 2 Supernova:
So what causes a dying star of said mass or greater to blow itself up to smitherines in such a way that more energy is released in the short span of the explosion that that of an entire galaxy for same time period. This is in contrast to the much more docile, gradual, and protracted conclusion of stars whose mass is less than 8xSun; planetary nebula after a red giant collapse and leaves white dwarf as remnant.
Massive Star Pre-Supernova(type 2) and implosion:
The core of a massive star will accumulate iron and heavier elements which are not exo-thermically fusible. Iron is the end of the exothermic fusion chain. Any fusion to heavier nuclei will be endothermic. Endothermic fusion absorbs energy from the surrounding layer causing it to cool down and condense around the core further. This added inward pressure is in addition to the inward pressure of gravity and causes a gradient where the core continues to collapse and get denser while exothermic fusion maintains expansive outward pressure above the gradient.
When the density of the inner core reaches a point where the pressure overcomes electron degeneration pressure. Then another gradient is formed where inside the gradient atoms no longer exists because the electrons of the atoms will join their nucleus. The protons in the nucleus combine with the electrons to become neutrons, and release neutrinos. There will be no more individual atoms inside of this gradient. The sudden collapse inside the electron degeneration pressure gradient of the core triggers the initial supernova implosion as said core collapses in a few seconds to a tiny point object of nuclear density (3×1017 kg/m3) no bigger than Manhattan.
The sudden in-rush of both the heavy element core outside the electron degeneration gradient, and the exotherimically fusible outer layers of the star cause immense heat and pressure in a few seconds period of time causing much of the lighter than iron elements to simultaneously fuse all at once.
Also, the extremely dense and energetic neutrino flux would be rushing outward from the mentioned EDP core collapse. When the neutrinos encounter head on the equally energetic atomic nuclei rushing inwards the following occurs:
When a neutron absorbs a neutrino, antimatter is formed as neutrons become anti-protons and positrons.
When this Anti-Matter encounters normal matter, we have the most efficient and complete mass to energy conversion possible:
Matter-AntiMatter Annihilation (MAMA).
The combination of the simultaneous fusion and the MAMA causes all hell to break loose in the greatest release of energy in the shortest time, 2nd only to the gamma ray burst, known in the universe.
What is left behind after most of the star has blown off is a solid extremely dense mass of neutrons around the size of Manhattan. Thus the term neutron star.
There is nothing denser than a neutron star. (See note below) It is just pure mass with no empty space left. To get a perspective what we are taking about here, our normal atom on earth is almost all empty space. If we take the simplest atom which is hydrogen, and the nucleus (one proton) was scaled up to the size of a pea sized marble in the middle of a football stadium, the orbital of the single electron would extend all the way at the outer edge of the stadium, the electron no bigger than a grain of sand; almost all empty space.
Now just imagine a sphere of solid mass of such marbles filling the stadium with no space left. That is how we can picture the density of a neutron star, and it is the density limit of the universe. A black hole has this same density, just more massive.
"A neutron star is so dense that one teaspoon (5 milliliters) of its material would have a mass over 5.5×1012 kg (that is 1100 tonnes per 1 nanolitre), about 900 times the mass of the Great Pyramid of Giza."
With the density limit constant, a neutron star can only get larger, occupy a greater volume, as it gains more mass. The mass and surface area determine the escape velocity, the velocity needed for an object to escape the host's gravity from the surface. The immense gravity causes the smallest neutron star to have an escape velocity of 100,000 km/s, or one third of the speed of light.
When the mass of the neutron star surpasses the level at which the escape velocity for the neutron star exceeds the speed of light, then even light cannot escape its gravitation, thus becoming a black hole.
Note: There is theoretically a point where the gravity and density of a neutron star will reach a threshold called the neutron degeneracy pressure. This was hypothesized by Tolman–Oppenheimer–Volkoff, and limit by same name. This would result in greater density called baryonic density. Then if we go denser, we can get quark density, and so on. These are all speculative theories since the sub-nuclear interactions are not well understood, and we can't look inside a black hole to see the evidence.