A gravitational singularity occurs when an astrophysical model, typically based on general relativity, predicts a point of infinite curvature. The term is closely related to the mathematical meaning of "singularity": a gravitational singularity occurs when the equations produce a mathematical singularity.
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 space-time are 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 non-rotating 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 Kerr black holes.
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 it is 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 vincinity 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, no theory of quantum gravity has been experimentally confirmed to date.
Shapiro, S. L., and Teukolsky, S. A.: Formation of Naked Singularities: The Violation of Cosmic Censorship, Phys. Rev. Lett. 66, 994-997 (1991)