In simplest terms, gravitational singularities happen when our mathematical models for gravity have a “divide by 0” in them. It is where our models for gravity break down.
The main place these occur is at the centre of black holes. But as “inside” a black hole is already a bit of a messy idea, it isn’t that big of a problem that our models break down at the centre of black holes.
The main thing that happens inside a black hole is you get really intense gravitational gradients, where the bottom of something is being pulled down much faster than the top, so things get stretched. Which is generally bad. Plus you get a lot of stuff being pulled in, so things are really hot and crowded/
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