To expand on what tiny-tim said, we say a singularity occurs when our laws of physics are no longer well defined. For example, in Newtonian physics, the gravitational force is expressed as:
When r=0, this equation is undefined (since division by zero is not defined), or we say that it is singular. I.e. there is a singularity at r=0. This is analogous to what happens with a black hole.