Dasymetric just means cookie-cutting the areas (like countries) for which you have data (like population) into more specific areas (NASA's zones of populated places) that do a better job of restricting those zones to where your data actually occurs.
One of the great things about dot density mapping is that it normalizes for area all by itself. You don't have to do a ratio of population divided by square miles (a cognitive abstraction) and then key that up to a sequence of colors (another cognitive abstraction) like you would with a choropleth map. But one of the sort-of drawbacks of dot density mapping is that the dots are randomly distributed within their areas -all you can control is how many dots. And if the dots are representing any sort of human data (like bicyclists, demographics, votes, commuters, downloads, etc.) then you are assuredly sprinkling many of them around in places that are obviously unpopulated. Sometimes that's ok, but sometimes it's pretty silly. Dasymetric mapping means we pare back our areas to only those that are likely to contain the phenomenon of interest (which was collected at that lower resolution).
Here's the Natural Earth cultural boundary file of countries. It is a fantastic and generous resource, which I hope you check out. It also has a field for estimated population, which I'll use as my example.
The countries of the world, provided by Natural Earth.
Inspired by Derek Watkins' fantastic squinty-eyed look at population densities (itself inspired by Wild Bill), I downloaded NASA's population density imagery and retained only the areas where they have estimated a population density of at least 5 people per square kilometer (seemed like a reasonable cut-off for "populated place"), converted it to a vector polygon set and then clipped the Natural Earth countries by it.
This is the result, in blue. You can download this file for your own pursuits, here.
The countries of the world, pared back to populated areas. Download this shapefile.
Before Dasymetric Clipping, and After
Here's a dot density map of the world showing population per country (each dot represents 50,000 people). By the way, I use population because it's convenient for illustration. So the map below is ok, and I do get a sense of which countries have more turf than people -overall, but it is not all that characteristic of how distributed the people are where people actually are.
Take Australia for instance. The average Australian lives in a city perched along the coast. And the fact that the rest of the continent is pretty much devoid of humans doesn't mean that they look around their neighborhood and see fewer people. It's an imprecise look at population for countries with vast stretches of low or unpopulated areas (i.e. pretty much everywhere except Europe). Alaska sucks up many of the dots that really ought to go into the lower 48. Other obvious examples of people dots happily dispersed in vast lonely tracts are Russia and Canada.
A standard dot density map of population (each dot represents 50,000 people). There is comfort in this view, because it is familiar, though that familiarity is a lie. Or at least wildly imprecise.
Below, I did the dot density mapping only inside my new clipped country shapes, where there is actually a population. Overall there is much less visual country-to-country variability in the density, compared to the map above, but this is a much more accurate* picture of real population density. Check out Australia. Now, instead of implying every neighbor lives miles away, I get a truer sense of their actual population density. I also get previously invisible insights into local distribution, like Egypt along the Nile, Canada in scattered cities nearer the border, Russia in their southwest, Algeria along the Mediterranean, and so on. Dots dutifully avoid deserts, water, tundra, jungle, and more deserts.
A sort-of-dasymetric dot density map of population (each dot still represents 50,000 people). While each country's overall population density (not all that meaningful, when you think about it) is harder to distinguish, the local population density is way truer. But maybe misleadingly precise?
*All of Cartography is a Lie
The act of constraining a dot density map to more realistic (dasymetric) zones has a couple of drawbacks. The first is that I lose that raw at-a-glance illustration of an overall national population density (though I would counter with the fact that this is a truer representation of actual local density). The second, and much bigger problem is emotional shenanigans. A dasymetric dot density map can impart too great an expectation of precision. Because I've taken the half-step of tighter population zones, it is entirely likely that readers assume more of its local distribution than is appropriate and take this is a direct and literal placement of people dots. A problem of scale and false confidence. The dot density map is still spraying dots randomly inside areas (it's just that the areas are more realistic). India is a good example of this. Because virtually all of India met the do people live here requirement (compared to scattered specks of population areas in the American west and pretty much all of Canada), all of India gets an evenly random dispersion of dots. In reality there is a way higher density of folks living in the northeast band. While this map doesn't have population data at that resolution, the precise speckles elsewhere in the world imply that it does.
The Cartographic Uncanny Valley
As a map-maker you will have to understand the balance of actual precision and perceived precision. Doing your best to take steps toward an honest representation of a phenomenon could mayyyybe land you in the cartographic truth version of the uncanny valley. Really generalized maps are clearly generalizations and are generously interpreted models of a phenomenon (at the cost of nuance). Really precise maps are understood to be more literal pictures of where (at the cost of larger trends and scale-ability). In-between precision can leave the reader wondering what level of scrutiny to apply which can inspire mistrust and revulsion. So there's that.
The uncanny valley. The hypothetical emotional response in terms of perceived realism. This concept may come into play when a map reader is unsure of the level of generalization a map is showing them.