Map projections
There are many different ways of drawing maps that represent the world's countries on paper. Since the countries are actually arranged on a sphere, any arrangement on flat paper involves some distortion of the areas and/or the relative locations of the countries.
Area proportional to population
The map areas of countries on some projections are very different from their land areas. This is particular evident in the Mercator projection where the size of countries in the far north and south are considerably exaggerated.
A different kind of distortion of areas results from intentionally altering the areas of countries to be proportional to their populations. There are many different ways to do this, while keeping roughly to the relative locations of countries and retaining their borders with other countries.
African populations
The map below shows all countries in Africa.
Click the checkbox Show circles to draw a circle centred on each country with area proportional to the land area of the country. Each circle contains virtually the same ink as was used on the original map.
Now select Area = Population from the pop-up menu at the bottom. The circles are now drawn with area proportional to the population in the countries.
Imagine these circles being moved and reshaped so that the circles for adjacent countries touch, but retain the same areas.
This distorted map would increase the map areas of countries with high population densities compared to the others.
Examples of world population maps
There is no unique way to distort the areas of the countries — there are several goals in such a map:
The map below places greatest emphasis on keeping recognisable shapes for the individual countries but their relative positions are distorted so much that some regions are almost unrecognisable (e.g. Africa and the Middle East).
The next map, produced by ODT Maps (http://ODTMaps.com) is much better, though the shapes of some countries (especially Russia) are quite distorted and the UK appears to be further north than Canada.
The map below is much simpler with each country represented by a rectangle, but it is still fairly effective for displaying the world's population.