For most, the Singularity is when artificial intelligence surpasses human intelligence. The consequences of this will create a future that is totally beyond human imagination. Machines more intelligent than humans will start designing machines that are more powerful than they are and processing power, which now increases exponentially, will explode.
For example. Charles Babbage once said that every single human conversation is recorded in the atmosphere. All you have to do is trace the movement of molecules back to the original sound. This is impossible, mostly because of the lack of computing power. The Singularity may change that.