In GR there are two types of singularity, actual and coordinate. This wasn't really figured out until the 1960's when people really began to get interested in black holes.

A physical singularity is a place in space-time where the density tends to become infinte, generally because the radius of the system goes to 0. We believe that to describe such things properly we will need new physics.

A coordinate singularity is a place that looks like a singularity in one set of coordinates, but that can be transformed away by using another set of coordinates. It is not physical. The event horizon is an example of a coordinate singularity.