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.
Sound scary? It is.
But - a key aspect of singularity theories includes the extension of human life and leisure. Work could become less a means of surviving, and more a way of expression as robots continue to replace jobs.
One promising - or dooming - aspect of the singularity is the possibility of humans living forever after having their consciousness "uploaded" into a computer system.
The exponential growth in technology is seen as the heralding sign of the approaching singularity as technology begins to meld within bodies eventually replacing biology with technology and brains with circuits meaning in the post-singularity world (if it can be called that) it will be impossible to distinguish between real and artificial intelligence.
While this may sound a bit sci-fi, our bodies and lives are already reliant on technology - for instance the watch is a technology which has been strapped to the body for centuries now. Smartphones can provide instant knowledge and advice to any environs or problems around us, and virtual reality is a wearable technology which disregards the body in an imagined space.
More practical considerations for the everyday include the prospect of world trade coming to a halt due to computational manufacturing (for instance, 3D printing) and food moving away from agricultural lands to laboratories.
While the singularity is strictly theory, these problems warrant serious thought from businesses and consumers as technology continues to accelerate and disrupt the economy and market with new advances.