In physics, a singularity is a point in spacetime where our currently developed theories are not valid anymore, we are literally not able to describe what happens inside, cos the density becomes infinite.
The technological Singularity, as described by Transhumanists, is a grade of technological development, where humans are not able to understand the undergoing process anymore. The technological environment starts to feed its own development in an feedback loop - computers help to build better computers, which helps to build better computers, that helps to build better computers...and so on.
So, when will the technological Singularity take off?
Considering the feedback loop, it is already present, maybe since the first computers were built.
Considering the density of information processing that exceeds human understanding, we may reached that point too.
Imagine a computer technique that is easy to set up and use, outperforms any humans in its task, but we can not really explain what happens inside, it is a black box.
Such an technique is present (and currently hyped) => ANNs, Artificial Neural Networks.
Of course we do know what happens inside, cos we built the machine, but when it comes to the question of reasoning, why the machine did this or that, we really have an black box in front of us.
So, humans already build better computers with the help of better computers, and humans use machines that outperform humans in an specific task and are not really able to reason its results....
obviously, +1 points for the Singularity to take off.