Population pyramids show population data by gender and age group. They are effective in showing changes over time. This chart of Japan demographics through 2050 appeared in The Economist:
This shows three pyramids: Male on left, right hand side females from 1950, 2005, and 2050. (2050 is projection based on current trends). From 1950 to 2005, the pyramid turned into a vase shape until 2050 when it is projected to be an inverted pyramid or wedge. What this visualizes is the aging of Japanese society — in a pyramid, the society has a large base of younger people. For the wedge, the base is inverted as the society ages, lives longer, and has lower fertility rates.
A variant of the chart was produced by the National Institute of Population and Social Security Research that shows three different population models for 2050, based on different fertility assumptions:
A time-lapse view of the chart is here.
An interesting view of the patterns of population growth is in this diagram:
This series of patterns shows population growth, stasis and decline.
One way I can think of improving the charts is to add a horizontal axis to show the male/female percentages by age band and also indicate average number of children by parental age by a color/shading within te band.