The diagram at the top of the Demographic page is a Lexis diagram, and it is the most useful picture in demography. Calendar year runs along the bottom, age up the side. A person is a diagonal line: born at the bottom in their birth year, moving up and to the right one year of age for every calendar year until the line stops.
Three ways of slicing the diagram correspond to three kinds of explanation for anything that happens to a population.
Age. A horizontal cut. Mortality at 80 is higher than at 40 because of what ageing does to bodies, whatever year it is and whoever the person is. Age effects are the most stable thing in mortality and the reason age standardisation exists.
Period. A vertical cut. Something happens in a calendar year that affects everyone alive at the time: a pandemic, a heatwave, a new treatment reaching the whole population, a change in how deaths are coded. Period effects show up as a vertical stripe across all ages at once. The pandemic is the clearest period effect in a century of data.
Cohort. A diagonal cut. People born in the same years share exposures that stay with them: the generation that took up smoking in the 1940s carried its lung cancer forward for fifty years; children born in a famine carry its consequences into old age; the cohorts born after about 1960 in high-income countries carry higher early-onset cancer incidence at every age so far. Cohort effects move diagonally across the diagram, and they are the ones most often mistaken for something else.
Why they cannot be fully separated. Age equals period minus cohort. Any of the three is determined by the other two, so a statistical model cannot estimate all three freely; some assumption has to be made, and different assumptions give different answers. This is the age-period-cohort identification problem, and it is why two competent papers can attribute the same rise in a disease to a cohort effect and to a period effect. When a study claims to have separated them, it is worth asking what it assumed.
Period life expectancy is not a forecast. The life expectancy quoted in the news — 79 for men, 83 for women, or whatever the latest year says — is a period measure: the length of life a hypothetical person would have if they experienced this year's death rates at every age for the rest of their life. Nobody does. A boy born this year will experience next year's rates at age one and the rates of the 2090s at age 70, and if mortality keeps improving those will be lower. Cohort life expectancy, which tries to account for that, is higher and is a projection; period life expectancy is a summary of the present. A fall in period life expectancy in a pandemic year does not mean anyone's life was shortened by that amount; it means that year's death rates were high.
What the diagram is for on this site. When an item reports a change, the first question is which cut it belongs to. A rise across all ages in one year is a period story; a rise that tracks birth year is a cohort story; a rise concentrated at older ages as a population ages may be no story at all. The diagram is the habit of asking.