By definition singularity is understood as a point in space-time at which the space-time curvature becomes infinite. And according to the general theory of relativity’s predictions there is a point in time at which the temperature, density and curvature of the Universe are all infinite, a situation which mathematicians refer as the singularity.
But perhaps the best way singularity is understood is through an acute perceptive of how black holes work. It is understood that during the dying stages, the cores of the massive stars (much bigger than our Sun) collapse spectacularly as the fusion reactions that used to power them at the first place dies down and gravity relentlessly pulls matter inward towards the star itself. Eventually out of millions of years of process, these stars collapses on it’s own self and completely disappear into a point that has been defined as one that is infinitely smaller than the full-stop we use to mark the end of a complete sentence. And this infinitely smaller than a full-stop comic point becomes a massively heavy, limitlessly dense and a dimensionless object known as singularity. And it is at this singularity that all matter in black hole is crushed out of existence. At this point it is important to mention that even though a singularity has absolutely no dimensions at all, however, black holes do have a size and the more massive the singularity is, the wider the region is around it which is, in fact, called event horizon and nothing can escape from the gravitational pull of the event horizon, not even light, and hence the name black hole.
It is believed that at singularity the laws of science and the mathematical power to predict the future break down and the singularities formed by gravitational collapse of stars occur only in places like black holes where they are decorously hidden from outside view by the event horizon. And this leads to the very fact of weak cosmic censorship hypothesis which states that this event horizon acts as a protecting (from visual perspective) mechanism for those who remain outside the black hole from the destructive consequences of the breakdown of the predictability that occurs at the singularity. According to experts of the singularity theorems of general relativity, there’s a big bang singularity that started the whole business of Universe, the tic of time itself, at the first place. And the singularity theorem research by noted experts like Roger Penrose and Stephen Hawking stress on the fact that the beginning of time had a point of infinite density and infinite curvature of space and time where all the known laws of science would break down, much like what would take place inside of a singularity within the event horizon of the black hole. It is understood that the Universe will continue to expand and at some point in the future it would collapse again (the Big Crunch) into a singularity like situation. And again even if the Universe does not go through a collapsing destiny, there still would be singularities in any localised regions that collapsed to form black holes. One interesting factor is that if time is measure in imaginary numbers (imaginary number is when a real number is multiplied by an imaginary unit), then we would find our Universe to be a finite and without any boundary as well as without any singularities.
It is understood that anything that falls into black hole is drawn inevitably toward the singularity at its centre which is an infinitely small point where all time, space, matter and energy end. But there just might be a way to avoid the singularity and instead pass through a small passage to emerge at the far end through a white hole which is also known as wormhole and into another universe or another part of our known universe. It is believed that if both ends of a worm hole are connected together with the aid of an electromagnetic or gravitational field, then it would be possible to keep entering and re-entering them to travel forwards and backwards in time, that means time-travel will become a great possibility.