Before Einstein's mathematical model, they were called "dark stars". Objects where the escape speed (determined by total mass and radius) is greater than the speed of light.
They can be small (for example, with a mass of "one Sun", called solar-mass black holes) or extremely big (the one at the centre of the Galaxy has a mass of millions of Sun), or anything in between.
The debate is whether certain objects can become black holes or not. For example, some of us think that once a big-enough supernova explodes (the explosion does not happen at the exact centre, but some distance from it), the iron core gets pushed inwards so hard by the explosion that matter will fall "into itself" so that even neutron degeneracy cannot stop the collapse. Since we know of nothing else that can stop it, we assume it "collapses inwards... forever". This means the radius of that mass goes down so close to zero that (from the outside) it would appear to have almost infinite density.
Others think that there may be other "layers" of degeneracy pressure that can stop it (for example, quark degeneracy pressure).
In any case, once it reaches a small enough radius that the escape speed is greater than the speed of light, no information from inside can ever reach us, so the discussion becomes irrelevant (for now).
The distance at which the escape speed is the speed of light is called the "event horizon" because anything "inside" of that can never reach us or even affect us. As far as the rest of the universe is concerned, it is AS IF the mass of the... "whatever" is located AT the event horizon.