A singularity means a point where some property is infinite. For example, at the center of a black hole, according to classical theory, the density is infinite (because a finite mass is compressed to a zero volume). Hence it is a singularity. Similarly, if you extrapolate the properties of the universe to the instant of the Big Bang, you will find that both the density and the temperature go to infinity, and so that also is a singularity. It must be stated that these come due to the breaking down of the classical theory.