This concept measures the ratio of working-age people to dependents (children and retired).  The idea is mainly that the dependents need a fairly constant amount of work (in the form of being raised, educated, etc.), thus they absorb the labour of a fraction of the working-age population.  The remainder of the working-age people are those who can perform in the roles traditionally investigated in economics, i.e. produce all the material goods and render the services that make up the wealth of the population.
Thus, if the ratio changes, it often has a disproportionately large effect on the ratio of wealth producers (as a fraction of the overall population), and as such, the standard of living (basically, the amount of wealth produced divided by the size of the whole population).
Roughly speaking, the three decades after the end of WW2 were periods of great economic growth across the developed world.  Up to the crises in the early '70s, the standard of living soared at an unprecedented rate, with yearly economic growth often in excess of 5%.
This huge growth of the economy was not, however, only intensive in nature (i.e. productivity gains), but also came with a significant expansion of the labor force (especially in the service sector).  This was counterbalanced for some time by other sectors (notably, agriculture) freeing up their own workers.  (This transition is visible as urbanisation.)  Eventually, however, the growing labor demand outstripped that made available by "conventional" means.
Starting in the early '60s, this meant pulling women into the "visible" labor force.  This simultaneously meant a reduction in the birthrate, for the very well-known difficulties inherent in having to work a wage job at the same time as caring for children and running a household.  This problem was further exacerbated by the trend of suburbanisation, which effectively ensured that most people would be living in households with only two adults.
This situation resulted in what I call demographic debt: a temporary rise in the demographic dividend achieved through decreasing the birthrate to sub-replacement levels.  The emphasis on temporary is important: this situation is inherently unsustainable.  After a generation's time has passed, the debt could either be paid back (which hasn't happened), or the generation taking on the debt has retired, or is on the verge of retiring.  At that point, the number of dependents is high, due to the large number of retirees, while the working-age population is comparatively low, because the age groups with low birthrates have now grown up.
Looking back to the first paragraph, we can see that the new generation is not only smaller in absolute numbers, but a larger fraction of the working-age population should get tied up in servicing the dependents.  However, this creates a very acute labor shortage.
Obviously, none of this is news.  The impending collapse of the retirement system has been a topic for at least a decade.  Northwestern Europe has, for several decades now, taken in huge numbers of immigrants (both from Southeast Europe and from overseas), mostly young adults, to make up for the lack of native working-age people.
I mentioned repaying the debt, and that quite obviously, it hasn't happened.  The other option is to default on it: in view of the labor shortage, even more people decide to miss out on having children, instead working for a wage.  This pressure only worsens the situation, much like unpaid interest compounding.  (The interest can be expressed in terms of the younger generations having to raise more children per capita, in inverse proportion to the fertility rates of the older generation (or the product of the rates, for multiple generations successively taking on such debt).)
Thus defaulting on the demographic debt means a collapse of the dividend, a consequent drop in the standard of living, accompanied by an acute labor shortage pulling in various people from abroad.  The recent flood of immigrants are here because we, Europeans, defaulted on our debt; they are here to foreclose.
To put the above in theatric terms, from the '60s on, the whole developed world made a deal with Nature: they borrowed an increased demographic dividend, a higher standard of living, for themselves, in return promising either to pay it back, or to forfeit their right to...
Now, I know the vast majority of people didn't understand this, and only acted from what they could see: wages and fashion.  Wages were high, living in suburbs was fashionable, and childless suburban two-adult families seemed ostentatiously wealthy.  Obviously everybody wanted to follow the trend.  And they didn't just seem wealthy, as I wrote two sentences ago, they genuinely were—but a good portion of it was borrowed from the future in some sense:
I know they didn't do this on purpose, but surely this should have been noticed and fixed by governments.  After all, this boils down to a very simple question: when you are retired, will your children and grandchildren be able to support you on the level you want?  If the answer falls short of "yes" on a societal scale, we have a problem.  Nature is going to collect on its deal.