If general relativity is correct, then the gravity inside of the event horizon of a black hole is enough to collapse all matter into a single point, a point where (I know this sounds hard to grasp, but bear with me) time itself comes to an end.
However, since not even light inside of an event horizon can ever get out (under relativity, not under quantum mechanics), we cannot see what is actually going on inside of a black hole. According to quantum mechanics, particles do not have a fixed position in space, and so might appear outside of a black hole spontaneously (this also sounds weird, and it is).
Some physicists think that gravity and space-time itself is subject to quantum mechanics, in which case this point also does not have a single position.
However, what theory adequately describes this, or if general relativity (which goes against quantum mechanics) is genuinely right, is still not known (and whoever finds out is going to get the Nobel Prize).
As for black holes, our mathematical theories predict that they have a number of really strange properties. Physicists would love to actually get a good look at a black hole to see if they do indeed have all the properties that we predict that they do.
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