Singularity is a state which witnesses "never seen before" technological progress somewhere after the invent of full-scale Artificial Intelligence.
Some people, like Prof. Robin Hanson, suggest that many singularities have occurred in the history during various technological achievements like agriculture, industrial revolution etc. But I think that all those events originated with some single technology achievement whereas singularity that we are talking about may comprise of dozens of such technology achievements like: robotics, AI itself, staggering power generation, outer space exploration, healthcare etc.
Singularity may or may not happen depending upon the source who invents the AI. If its some government then most likely its going to remain a secret and mainly used for military purposes. If its done by some company then it can commercialize the newly invented technologies thus bringing singularity closer.
Another interesting thing to notice here is that how the AI will be used. Will it be allowed to put its resources on developing powerful technologies or will it be used to provide services, solve mathematical problems, analyze scientific data and so on. Also there will be many powerful entities trying to control the AI and its outcomes.
So, singularity is the state which may or may not happen as the world is just too complex.