A singularity is a point, usually thought to sit at the center of a black hole, where the equations of general relativity predict that matter is crushed into infinite density and curvature. Space and time as we understand them stop making sense at this point, and the theory simply cannot describe what is actually happening there.
Most physicists do not think an actual infinity exists in nature. Instead, the singularity is usually seen as a sign that general relativity is incomplete, and that a full theory of quantum gravity, one that properly merges general relativity with quantum mechanics, would replace that infinity with something finite and physically sensible.
The Big Bang is sometimes described the same way, as a singularity where the entire observable universe was compressed into an infinitely dense point. As with black holes, this is generally read as the limit of what our current theories can describe, rather than a literal, physical infinity that existed.
In the 1960s, Roger Penrose and Stephen Hawking proved mathematically that singularities are not just an artifact of simplified, idealized models, they are an essentially unavoidable consequence of general relativity whenever enough mass collapses under its own gravity, work that earned Penrose a share of the 2020 Nobel Prize in Physics. Penrose also proposed something called the cosmic censorship hypothesis, the idea that singularities should always be hidden safely behind an event horizon rather than exposed to the outside universe as a so-called naked singularity, which physicists worry could break down predictability entirely. Whether cosmic censorship actually holds in every possible situation remains an open mathematical question, and it is one of the reasons singularities sit at the very edge of what general relativity alone can be trusted to describe.
Not every singularity is expected to be a simple point. The mathematics describing a rotating black hole predicts a ring-shaped singularity instead, an unusual geometric result that has led some theorists to speculate, largely as a mathematical curiosity, about exotic possibilities like traversable wormholes, though nothing resembling a safe, usable passage through one has ever been shown to actually exist.