Gravitational singularity
From Academic Kids

A gravitational singularity occurs when an astrophysical model, typically based on general relativity, predicts some type of pathological behavior of spacetime, such as a point of infinite spacetime curvature. The term is closely related to the mathematical meaning of "singularity": a gravitational singularity occurs when the equations produce a mathematical singularity.
The notion of singularities as points where spacetime curvature "blows up" is the one that is most intuitive, however, singularities can exist even if the curvature of the spacetime is finite everywhere.
More generally, a spacetime is considered singular if:
 It is geodesically incomplete, meaning that there are freelyfalling observers whose existence is finite in at least one direction of time ( as measured by their local clocks ). For example, any observer below the event horizon of the nonrotating black hole would fall into its center within a finite period of time, at which moment laws of physics would break down and it would become impossible to predict the observer's further evolution. Thus, we say that there is a gravitational singularity in the center of the black hole.
 Spacetime also has to be inextendible, i.e. not to be a proper subset of some bigger spacetime. It's fairly easy to construct spacetimes that possess incomplete geodesics from regular Minkowski space by removing points, yet we want to avoid calling such constructs 'singularities'. See Rindler coordinates for a fairly involved example where an apparent singluarity arises because of cutting a wedge out of Minkowski space, followed by coordinate transformation.
If these two conditions are met, it is said that singularities are located at the "points" where "incomplete" observers start and/or end their existence.
The Big Bang cosmological model of the universe contains a gravitational singularity at the start of time (t=0). At the "Big Bang Singularity," the model predicts that the density of the universe and the curvature of spacetime are paradoxically infinite. However, the basic Big Bang model does not include quantum effects, so its predictions are valid only shortly after the projected singularity.
A singularity also exists within a black hole, where general relativity predicts a region of infinite curvature. In a nonrotating black hole, the singularity occurs at a single point in the model coordinates, and is called a "point singularity". In a rotating black hole, the singularity occurs on a ring, and is called a "ring singularity". Rotating black holes are sometimes referred to as a Kerr black hole.
Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship principle. However, in 1991 Shapiro and Teukolsky performed computer simulations of a rotating plane of dust which indicated that general relativity allows for naked singularities. What these objects would actually look like is unknown. Nor is it known if singularities would still arise if the simplifying assumptions used to make the simulation tractable were removed.
Many physicists believe that gravitational singularities are "unphysical", meaning that general relativity ultimately ceases to be an accurate description of gravity somewhere in the vicinity of what would otherwise be a singularity. It is generally assumed that a theory of quantum gravity  a theory that unifies general relativity with quantum mechanics  will provide a better description of what actually occurs where general relativity predicts a singularity. However, as of 2005, no theory of quantum gravity has been experimentally confirmed.
References
 Wald, R. M.: General Relativity, University Of Chicago Press (1984)
 Shapiro, S. L., and Teukolsky, S. A.: Formation of Naked Singularities: The Violation of Cosmic Censorship, Phys. Rev. Lett. 66, 994997 (1991)de:Singularität (Astronomie)
nl:singulariteit fr:Singularité he:סינגולריות כבידתית hu:Gravitációs szingularitás pt:Singularidade gravitacional sl:gravitacijska singularnost th:ภาวะเอกฐานเชิงความโน้มถ่วง fi:singulariteetti