Between 1965 and 1970, Penrose and Hawking proved a set of three singularity theorems together, all of which make varied assumptions and have different starting points. ** when a cosmologist refers to these as the "Penrose-Hawking Singularity theorem" they're mostly interested in the last theorem, which was proven in 1969 in a famous paper titled "The singularities of gravitational collapse and cosmology". This is the most powerful formulation of the Penrose-Hawking theorem, and it's what I'll talk about here.
Defining what a singularity is can be difficult. In some places you may find that a singularity appears in a model when the null or timelike geodesics cannot be extended indefinitely; that is to say, if the paths of photons or particles with mass terminate at some finite time into the past. Penrose-Hawking try to show that this condition obtain in our universe given some general features of spacetime.
Physically, interpreting that would mean that the universe had a beginning (or local regions of spacetime end in black holes). If the universe began with a singularity we would have an absolute beginning of the universe from nothing, not just empty space or a vacuum because it is impossible to extend a geodesic beyond a singularity. George Ellis describes it like this, pg. 5 "This is not merely a start to matter — it is a start to space, to time, to physics itself. It is the most dramatic event in the history of the universe: it is the start of existence of everything."
Non-technical, simple explanation
A singularity will appear in the past, if the following five conditions are met:
(a) There are no closed time-like curves in our past. ***
(b) The Universe contains enough matter to create a closed trapped surface.
(c) Generic energy condition (geodesics encounter curvature in our past, that isn't specially aligned with it).
(d) Strong energy condition (gravity is always an attractive force).
(e) Our spacetime satisfies the equations of Einstein's General Theory of Relativity.
You can construct models that violate any one of these assumptions, for example Gott and Li have a model where the universe is a time machine, that violates (a). Option (b) could be violated in some exotic spacetimes if the universe perfectly counter balances the effects of gravity, with some de-focusing characteristics. Option (c) could also be violated in some physically unreasonable models, that are very specially constructed. These possibilities aren't taken very seriously, though.
Most cosmologists expect the Penrose-Hawking theorem to fail, either on option (d) or (e). Either because General Relativity will be replaced with a quantum theory of gravity, that smears out the singularity with quantum effects (i.e., they add some 'quantum term' to the second Friedman equation, from their favorite candidate theory). Or, because the strong energy condition is violated in quantum fields which include a "negative pressure vacuum". As Penrose and Hawking state "our theorem cannot be directly applied when a positive cosmological constant is present". That is, where the pressure is less than minus one third of the density:
Energy conditions like this are normally formulated using the energy-momentum tensor, however I've written it here using pressure and density, under the assumption that the universe behaves as a perfect fluid.
Slightly more technical explanation
Penrose-Hawking use a null and timelike version of the Raychaudhuri equation for geodesics that don't rotate and satisfy the strong energy condition, implying that they must intersect at some finite proper time. The Raychaudhuri equation is. ****
This is the basic equation of gravitational attraction for cosmology, it tells the cosmologist how the curvature of space along each geodesic or world line is determined by the energy density and pressure. You can integrate this, along with energy conservation to get the first Friedman equation.
More importantly, the equation has a term for the expansion, the shear scalar and the curvature. So that it describes if a congruence of geodesics focus or defocus, if each of these terms are negative, it tells us that if a set of geodesics start out focused they will evolve and have continued focusing, and that spacetime will be geodesically incomplete.
Each of the assumptions Penrose-Hawking made is an assumption either about initial conditions which tell us the geodesics start out focused, energy conditions which tell us they continue being focused or causality conditions which tell us there are no focal points. A spacetime which satisfies these conditions will be geodesically incomplete into the past.
Stephen Hawking changed his mind
Very famously, on page 50. of a Brief History of Time states "there was in fact no singularity at the beginning of the universe". In the Hartle-Hawking no Boundary proposal the initial state of the universe is treated with imaginary time values for the time parameter in Einstein's equations, instead of real time values. This treats time like a dimension of space, and the only "beginning" in the model occurs in this region where there is no well defined distinction between past and future. It thus becomes arbitrary which 'point' you pick as the beginning of your spacetime.
* In 1965 Roger Penrose showed in a paper titled "Gravitational collapse and spacetime singularities" that once a star collapsed passed the point where r = 2m it would not come out again. This result was already suggested by the Schwarzschild solution to GR field equations, but this involved integrating over a perfectly symmetrical sphere. Even small amounts of angular momentum in Newtonian mechanics allowed that solution to be violated, and the star could re-expand. However, Penrose's proof introduced the concept of a closed trapped surface, a region where light and matter cannot escape due to the intensity of gravitational forces and was thereby able to drop the assumption of exact symmetry.
** Hawking and Ellis were inspired by Penrose's 1965 paper and wrote in a letter to the editor a new singularity theorem generalizing the work of another cosmologist, Shepley from 1964 who worked on homogeneous models.
*** Hawking was able to prove a singularity theorem three years earlier, in 1966 which made no assumption about closed time like curves but this theorem only applied to closed, 'everywhere expanding' models of the universe, that possessed a certain geometry. This was then replaced with the 'closed trapped surface' assumption.
*** Hawking showed in 1992, that in classic theory a CTC would destroy itself, forming a singularity. In other words CTCs violate what became known as the "Chronology Protection Conjecture". Though this assumption is no longer necessary, more modern singularity theorems do not have a 'no-CTC' assumption.
**** This won't be how you'll find the equation written virtually anywhere (the proper derivation should include tensors) but I've tried to keep it relatively simple. This equation is also known as the "acceleration equation" and I've written about it here.