Also interesting to note, were the sun to collapse into a black hole this instant, nothing would change on earth (other than the lights going out after eight minutes and everyone freezing to death after some time). All the planets of our solar system would still orbit the black hole as they did before with the sun, because the amount of mass, and with that also the amount of gravity given a certain distance, did not change.
The thing that makes black holes or very dense mass suck so much is that you can simply get much closer to the center of the gravity well, and by concentrating all the mass on a small spot you also make it go "deeper." Of course this doesn't mean the gravity would increase if you were to hypothetically dig down to the center of earth - at that point it would actually feel like zero-G again, because now all the mass is around you, pulling you equally in all directions.