Okay, there are lots of meanings for singularity throughout maths, science and technology, so we'll focus on the one found at the centre of a black hole, the gravitational singularity. When scientists refer to the singularity at the centre of a black hole, they're describing the point at which the density of the black hole approaches infinity. Basically, it has infinite mass in an infinitely small, one-dimensional space. The singularity is the point in the universe at which the laws of physics, as we know them, completely break down. As both space and time, the two things that pretty much make up our entire reality, essentially cease to exist at the point of singularity, it makes them impossible for us to "figure out" using our current understanding of physics. The term "singularity" is a little bit like "dark matter" in the sense that it is used as a bit of a cheat in maths, and we don't actually know what one is, or if they even exist at all. In fact, you may not even hear the term "singularity" thrown around for much longer as some theories of quantum gravity have done away with the singularity altogether.
