Dr. Stephen Hawking is attempting to unite the theories of the very large (Einstein's theory of general relativity) and theories of the very small (quantum mechanics). A comprehensive theory of everything. Why are we here? Where did we come from?
Einstein's theory of relativity applies to large objects: planets, the sun, stars, the galaxy, the universe. The theory predicts the existence of black holes. Black holes are objects in space of infinite density, not even light can escape. Roger Penrose used Einstein's equations to prove that at the center of a black hole is a "plug-hole" - a "drain" into which space, time, the entire universe disappears. Not even light can escape, hence the name "black hole." This "plug-hole" is called a singlarity and is effectively infinitely small. Time and space come to an end inside a singularity. The drain of death for the universe. As Michio Kaku noted, the word singularity simply hides our ignorance: we do not know what the singularity actually is in a physical equation sense.
Equations have inverses. For addition there is subtraction. For multiplication there is division. Hawking took Penrose's equations and reversed the time component - he inverted the equations. And proved that the universe must have arisen from a singularity. The catch was that the singularity remained unexplained.
The singularity arose as a result from Einstein's equations. But the singularity is infinitely small. Einstein's equations predicted it could exist, but could not describe the singularity. Only the theory of quantum mechanics works when things are infinitely small. But quantum mechanics does not currently explain gravity. Quantum mechanics can explain mass, electrons, protons, electricity, magnetism, but not gravity. A theory of quantum gravity has yet to be developed. Such a theory would have to yield Einsteinian results at larger scales such as planetary or galactic scales.
Everyone was stumped. They needed a place in space where both quantum mechanics and relativity equations could be generated and solved. Getting into bed one night, Hawking realized that the place both equations could be solved was at the edge of black hole. At the very edge the equations could be solved.
Hawking made an amazing discovery. Quantum theory had already proved that in empty space particle-antiparticle pairs are formed and then annihilate. The appearance and disappearance of particle pairs affects the spacetime fabric of space and hence the effect is also referred to as quantum foam.
Next to a black hole these quantum particle-antiparticle pairs get separated with the antiparticle falling into the black hole. This particle has antimass, negative mass, with the result that the black hole loses mass. Shrinks. The resulting energy emitted by the other particle is now called Hawking radiation.
This small step was the first time quantum mechanics and general relativity had been joined to make a prediction. This was a starting place for uniting the theories. Hawking had thought that further results would flow quickly and easily. In this he was not correct. Further progress has been made, but with difficulty.
Data gathered by the WMAP (Wilkinson Microwave Anisotropy Probe) provides evidence that the singularity was not smooth and uniform - the background radiation in space left over from the big bang is not smoothly distributed. This confirms that quantum fluctuations and uncertainties were at work in the early universe. This indicates that quantum theory is the appropriate theory for moving forward on describing the singularity. An imperfect start led to the formation of galaxies, stars, and empty space. Had the start not been imperfect, then planets, stars, and galaxies would never have formed. Imperfection is the key to the existence of the universe.