In old-form general relativity, a singularity would be a flaw in space-time, a discontinuity, a location of infinite density and space-time curvature. Black holes cloak that singularity in an event horizon (which should form before the singularity does during stellar collapse) such that, despite all the gross violations of physical law that the singularity seems to imply, none of it matters because it cannot effect the outside universe. Nothing that happens inside the event horizon can influence anything outside. Break all the rules you want inside that horizon; no harm, no foul. That's usually referred to as the Cosmic Censorship Hypothesis.
But, if the singularity is naked, if there is no event horizon... Well, that's not supposed to happen. Some quantum gravity theories say it could happen, though. And I think there may even be a solution... yeah, pretty sure, there is a solution in old-form general relativity where, if the black hole is spinning fast enough, the event horizons collapse and the singularity is exposed. Depending on what (if any) modern theories of gravitation you buy into, that can be ok (it wouldn't really
be a singularity, density would be unable to truly become infinite) or it could be very very
WTF. I mean, what you basically have is a place where the universe ends, and it is exposed for all the world to see.
Man, I'm dying to know where they are taking this.