A gravitational singularity (sometimes called a spacetime singularity) is a term used to describe the centre of a black hole where the gravity is thought to be infinite. However, most scientists do not think that it is actually infinite, but it is useful for their maths to treat it like it is.
In the center of each black hole is a singularity, a point of zero size and infinite density. The singularity is the strongest and deadliest part of a black hole. This is indeed difficult to grasp. Actually at the center of a black hole spacetime has infinite curvature and matter is crushed to infinite density under the pull of infinite gravity. At a singularity, space and time cease to exist as we know them. The laws of physics as we know them break down at a singularity, so it's not really possible to envision something with infinite density and zero volume.