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The point at which humans create something more intelligent than themselves.

The Singularity, as a technological concept, has been known about for longer than people realize. In 1958 Stanislaw Ulam talked about it as

...the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.

This actually describes colossal change without making any assumptions regarding that change being good or bad for the human race as a whole. A simpler definition of the Singularity is the point at which humans create something more intelligent than themselves. The significance of this is that the intelligence would then be able to make itself even more intelligent, having surpassed humanities best effort. In so doing, a runaway cycle begins with each iteration bettering the previous one.

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The 4th industrial revolution

Also being talked of as "Industry 4.0", the rapid convergence of sensor data from the IoT explosion with powerful cloud based AI means the next paradigm shift is already underway.

1Steam

Using energy from steam and water for mechanical production. 

2 Mass production

Division of labour + component specialisation = assembly lines.

3Electronics

Computers, IT, automation and mass communication.

4AI

Automated descision making and automated manufacturing processes via robotics.

The evidence of the dramatic change bringing about this AI revolution is all around us - self driving cars, robotics, genetic engineering, superfast, superconnected devices. We are already familiar with the mechanisms Ai will use to get smarter - it's nothing more than the software upgrades we all do all the time anyway. When your smartphone alerts you to a new version of an app, you don't care how that improvement was created, just that you assume it is better than the previous one, so install it. You also expect there to be more in the future - it is exactly as if you are watching digital evolution in the palm of your hand.

Kurzweil

The best known work in the field is Ray Kurzweil's The Singularity is Near. He actually dates it at 2045, "a point where progress is so rapid it outstrips humans' ability to comprehend it.". 

Will it ever be possible?

There is, however, some contoversy regarding the whole idea of the Singularty. Danko Nokolic is a neuroscientist at the Max Planck Institute for Brain Research in Frankfurt. He boldly claims "You cannot exceed human intelligence, ever. You can asymptotically approach it, but you cannot exceed it.” He argues genes are equally important to the role of intelligence in humans, and they cannot be simulated using current technology to create thinking machines.

The state of play right now is that many separate, specialised AI-like systems are being developed to perform distinct roles, such asthe mapping example mentioned earlier, manufacturing control, finance etc. Once these systems evolve towards a single central processing unit, capable of carrying out all their previously separate role, you will be much closer to the Singularity than ever.