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.