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.