I recently came to understand the singularity, and now that I do the argument is pretty funny; it basically is the result of a bad mathematical model.
Here's the idea: Suppose you built a computer to design a new computer twice as fast as itself, whose job is the same, and so on for its children, ad infinitum.
The first computer takes 1 day to do its job. Its speed is 1 exaflop.
The second computer takes 1/2 day to do its job. Its speed is 2 exaflops.
The third computer takes 1/4 day to do its job. Its speed is 4 exaflops.
...and so on.
Hence the total time it takes all the (infinite) computers in this series to work is given by the convergent geometric series,
1 + 1/2 + 1/4 + 1/8 + ... = 2 days.
In other words, you design an infinite number of computers in finite time. And in this finite time, you've managed to produce infinitely fast computers, since their speeds go with 2^N. So the event referred to as "the singularity" is not called this just out of a love for words that sound like they belong in bad Star Trek dialogue; rather it is literally a mathematical singularity of the day-number--to-computer-speed map.
Kind of silly, no?