What is the singularity?
Put simply, it's the point at which technology and artificial intelligence surpasses human capability and brainpower.
The term "singularity" originates from the usage in physics to refer to the point at which a black hole's mass becomes infinitely small and space-time curves infinitely while gravity is infinitely strong and anything which enters is crushed into an infinite density and the laws of physics have no application.
A technological singularity is the point at which artificial intelligence reaches a point at which it can self-upgrade and rewrite software to advance quickly beyond the realms of human knowledge.
Basically, machines reach a point in which they can upgrade and improve exponentially. Because humans think in a linear fashion, it's innately difficult to comprehend exponential growth. Therefore machines could upgrade and progress at a literally incomprehensible way.