A black hole is thought to be an object whose gravity is so massive that the Newtonian laws which normally apply are out the window. The gravity is so huge that even light cannot escape, hence the name black hole.
Even weirder things are observable near a black hole: at a certain point above the object, an observer looking along a line perpendicular to its surface would see the back of their own head in front of them, thanks to light rays curving back upon themselves. In short, Sir Isaac's rulebook doesn't apply at all: you need a whole new set of laws, and this is what a singularity is all about.
Vinge posited something he called the technological singularity in a 1993 paper he prepared for a Nasa symposium on the future. He wrote about what might happen if super-intelligent machines could be created, and figured that a really smart machine could design and build even smarter machines, which could build smarter ones, and so on.
The result might be that the world's technology would begin to change so rapidly that human intelligence couldn't keep up. We would no longer be able to peer confidently even a short distance into the future. Things we thought would take a million years would arrive in a century, and other things could change in mere hours.
Personally, I think we're pretty close to the singularity already, even without super machine intelligence. Technology is moving so fast that we can no longer reliably predict the future even a few years ahead. To which I would add that we may also see things that are pretty weird - like cloned animals and teenage hacker gods - a la the black hole's curving light.
For much of history, people have been able to make pretty good guesses about the future, even the relatively distant future - say, five generations, 100 years ahead. For example, last weekend I was walking to a hotel through downtown Atlanta's infamous Olympic Centennial park, thinking about how I'd gotten there from my home near San Francisco. I'd driven to San Francisco airport, hopped a jet to Atlanta's Hartsfield Airport, where I took the robotically-controlled tram from the gate to a Marta train (Atlanta's underground). The journey had taken eight hours door-to-door.
A person in the year 1700, if asked to predict how a person in 1800 might travel from where San Francisco would someday be built to the spot where Atlanta now stands might have said by horse. Their only wonder would be why someone would undertake such a difficult and dangerous journey. James Watt's first practical steam engines were still some 70 years in the future, and the Industrial Revolution was yet to be felt. My journey would have taken perhaps three months in those days, as it would have for the preceding millennium.
A person in 1800 might have said the same if asked to forecast long-distance travel in 1900. The Industrial Revolution had already profoundly changed life at the beginning of the 19th century, but the first railroad was still 30 years in the future. Horses would have been a likely and accurate prediction for 1900.
But a person in 1900 would have had a hard time predicting accurately how I'd gotten from California to Georgia for all the technological wonders that the 19th century had wrought. Perhaps influenced by Jules Verne, she might have predicted a very fast, futuristic train of some sort, getting the short part of my journey right. But to get it all right, she'd have had to foresee the automobile, the jet and robotics from a time before the Wright Brothers flew, Henry Ford innovated or the word "robot" was coined.
This century has seen more technological change than all of human history before it. Current technologies beget new technologies which beget yet newer ones. Change is so certain to accelerate that we've even codified it in Moore's Law.
Industries in the 1950s turned to the science of statistics to manage quality and forecast demand. Computer companies in the 1990s have turned to just-in-time manufacturing because it's too hard to foresee what the computer market will be like in the 18 months or so it takes to design and market a new machine.
Technologies change so rapidly, and new companies are needed so often, that a whole industry, venture capital, profits mightily by fostering them. The media that have sprung up to describe this new world change their vision frequently (as noted in Wired magazine's monthly "Tired" and "Wired" lists).
Hundreds of existing technologies can combine in millions or billions of ways - a so-called combinatorial explosion. There is not time enough in the universe to consider each of the possibilities, yet the future's wonders lie among them.
So what are we in the technology biz to do? Good technologists are valued at least in part because of their crystal ball skills. Nothing is certain but death and taxes - and ever more rapid change.