The Constant-Fertility Variant mechanically assumes that fertility rates will remain right where they are. If you're at 1.1 kids per woman, it assumes that you'll always have 1.1 kids per woman.
The other approaches do something quite different: With a few provisos, they assume that countries' Total Fertility Rates (TFRs) will asymptote to a specific rate. In the Medium scenario, the rate is 1.85 children per woman (it used to be the "replacement rate" of 2.1). If this sounds weird, it is. As Wattenberg explains:
Until the publication of the 2002 data, the UNPD "Medium" scenario took every modern nation with a TFR below 2.1 children per woman and by statistical fiat sloped it upward toward 2.1 by 2050. Nations above the replacement rate were sloped downward to 2.1.
Now the UN projects that TFR will fall to 1.85. Recall that every single developed nation - all of Europe, Japan, the United States, Canada, Australia - is currently in subreplacement status, and almost all of them have been there for decades. Yet for the most modern countries the new UN projections will raise the TFR for the next fifty year cycle.
In other words:
[F]or nations with low TFRs, this does not mean a decrease in fertility, but rather a large increase. Thus the current German TFR is 1.35. Using the new level of 1.85 means going up by half a child per woman, which is quite substantial. As we shall see, there is no evidence that such an increase is occuring.
The High and Low projections basically use the same methodology as the Medium, but asymptote TFRs to different numbers: the High assumes TFR will go to 2.35 children per woman; the Low assumes it will go to 1.35 children per woman. Thus, for many countries like Russia, Japan, China, and Italy, even the Low projection actually predicts a reversal of a decades-long fertility decline.
To get some perspective, suppose that you're trying to predict CD sales in the developed world in 2050. You could assume that sales per person will continue at their current rate. That's like the Constant-Fertility Variant. You could assume that sales per person will asymptote to a level somewhat above the current rate. That's like the Medium Fertility Variant. You could assume that sales per person will asymptote to a level far above the current rate. That's like the High Fertility Variant. And you could assume that sales per person will asymptote to a level somewhat below the current rate - even though large segments of the market are already well below that rate. That's like the Low Fertility Variant.
Now notice: The most obvious projection - one in which the decline in CD purchases continues - is conspicuously absent! And at least for CDs, isn't that the most reasonable projection? CD sales per person will fall to near-zero, and the world stock of CDs will slowly do the same.
Are people like CDs? At least for the near-term, there's every reason to think so. If fertility has been declining for decades, it's strange to assume that fertility coincidentally bottomed out yesterday. So Constant-Fertility Projections are actually probably on the high side.
In the longer-run, though, evolution will almost surely save us. If the average woman has one child, population size shrinks by 50% per generation. But if 10% of women have three kids, and if family size (like virtually every trait) is partly heritable, the proportion of the population that wants 3 kids will exceed 50% in a few generations. In a century or two the desolate villages of Italy will be reclaimed by the descendants of those of us who think that life is a chain letter worth continuing.