To understand singularities, you have to think about infinite density, like how the original Big Bang started. The Big Bang, most scientists believe, started from an earlier universe that was pulled together by gravity into a single point. After that, it exploded and made all the matter in the expanding universe. A black hole singularity is similar, except on a smaller scale. Imagine if our sun decreased to the size of the nucleus of an atom. It would be very dense. However, our sun does not have a high enough mass to become a black hole. It is well below the Chandrasekhar limit, which means it will shrink to a white dwarf, with radius of a few thousand miles. It can't become a black hole, because it is too small.