A star, that will go supernova, apparently, can become a black hole, or a neutron star; how does that work?

I think it is a matter of mass.
When a supermassive star dies, the fusion reaction becomes too weak to sustain the star’s size, so the star collapses in on itself, like a balloon that is deflating.
At the end of that collapse, it increases the pressure in the core, causing a secondary fusion event, which both causes the outer material to be explosively expelled, which is the supernova, and the core to be further compressed into either a core made almost entirely of neutrons, or, if the collapse and resulting explosion was strong enough, into a black hole.
But since you want me to pretend that you are my grandfather:
What do you care, gramps, you will be dead soon!

@ragingloli is correct. The magic number is called the Chandrasekhar Limit. Anything under that limit becomes a stable white dwarf (like our sun). Anything over that limit will theoretically collapse to a black hole or neutron star.
https://www.britannica.com/science/Chandrasekhar-limit

Yeah. What they said. For once the size counts.

Actually, in reading my answer it implies that our sun is a stable white dwarf. It is, of course, not a white dwarf—it is a yellow main sequence star. But after the star goes nova (it will not go supernova as it’s too small) it will collapse to a white dwarf and then ultimately burn out. It will not become a black hole because it is not massive enough.

Like I said, size matters.

I think I have the extra resources to take another college astronomy class. I think I will do that.

@Caravanfan our sun will be a stable white dwarf. Like our president is.

Thanks, guys, for your insight, and the interesting reading material.