If you don't know what the singularity is - and you have some time - read the following article by Ray Kurzweil,
The Law of Accelerating Returns. It's one of my favorite articles and informs much of my thinking of the future of technology.
Processing speed is growing exponentially. While the first few doubles may not be noticed, or seem exciting to outside observers, it can soon grow to game-changing proportions.
I got the above table from Intel, published in 2005. Six years later the Core i7 (Sandy Bridge E) has 2270 million transistors (2,270,000,000).
Where will we be in 20 years? Assuming computing power (equivalent but not equal to transistors) continues to double every 18 months we will go through 13 more doubles, 2.27 billion will become 4.5 billion in 18 months and become 9000 billion in 20 years. Imagine that if you can.
Twenty years ago we had clunky "portable phones" with huge batteries, today we carry around TVs in our pocket. Tomorrow will it carry all our books, medical records, photos? Of course. We need to imagine what new things could be there. Could it be a portable "pensieve?" A place to store and review memories?