The other answers here are pretty good, but I feel like there's a point that needs to be noted, which is that we have very limited data about black holes or comparably energetic phenomena, and our understanding of the physics involved in them is thus also limited.

Newtonian physics is a pretty good model of how things operate. It can explain why levers make things easier to lift, why the planets in our solar system move the way they do, how an internal combustion engine works. Virtually everything you encounter in everyday life can adequately be handled by Newtonian physics.

But it's not perfect. Things that are moving really fast, or are really hot or really dense, do things that Newtonian physics doesn't predict. Light was the first thing people noticed behaved weirdly; it moves so fast that apparently it can't be sped up, even by things as fast as the rotation of the earth. This was the observation that led Einstein to invent General Relativity, which is a stunning piece of extrapolation. General Relativity is a complex theory, which, at low energies, gives exactly the same results as Newtonian mechanics; but for very fast objects, or very massive objects, it handles things differently. This is where the imagery of spacetime as a stretchy sheet comes from; it's also where the idea of light being bent by gravity comes from, and the very idea of "spacetime" itself, in which time is treated as impossible to separate from space, and equally malleable. Thanks to better telescopes and computers and astoundingly accurate clocks, we've been able to verify some of the counterintuitive predictions of relativity--we've observed light being bent by gravity, in the form of "gravitational lensing" as a galaxy passes between us and a star; our GPS satellites have to correct their clocks for the fact that time moves a tiny bit slower when you're orbiting the planet at high speed; and we've made use of the theory's predicted equivalence between mass and energy in the form of nuclear reactors that provide electricity. So we know that General Relativity is a better theory of gravity than Newtonian Mechanics.

At the same time, there's another theory that's also developed, which is Quantum Mechanics. Quantum Mechanics also makes different predictions than Newtonian Mechanics, but it makes those predictions at the opposite end of the spectrum--it's the theory of the infinitesimal. It's why negatively charged electrons keep orbiting the positively-charged nucleus of an atom, instead of being dragged into it. Now, it's been verified extensively too--it's actually a pretty everyday thing now, since manipulating the tiny and the low-energy is quite useful. It's used for things like lasers, which are seriously everywhere now. The gyroscope in your smartphone relies on lasers.

The problem with this is that quantum mechanics doesn't allow for the manipulation of spacetime. Space and time are implicitly constant in quantum mechanics, just like they are in Newtonian mechanics. So quantum mechanics and general relativity are fundamentally at odds with one another.

This doesn't stop us using them for quite a lot of things, because, in our everyday experience, the two don't tend to interact. The quantum stuff is for things that are extraordinarily tiny, and the relativity stuff is for things that are extraordinarily energetic or massive, and there's not a lot of overlap in those two domains. In between, the strange aspects of both fade away and tend towards Newtonian physics.