The notion of singularity
In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of double points, at which the string crosses itself in an approximate 'χ' shape. These are the simplest kinds of singularity. Perhaps the string will also touch itself, coming into contact with itself without crossing, like an underlined 'U'. This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the 'U' away from the '_'.
How singularities may arise
In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by degeneration of manifold structure. That implies the breakdown of parametrization of points; it is prominent in general relativity, where a gravitational singularity, at which the gravitational field is strong enough to change the very structure of space-time, is identified with a black hole. In a less dramatic fashion, the presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired 'corners' in a process of folding up resembling the creasing of a table napkin.