What is the half-life concept used to measure?

Radioactive decay - the time for half the atoms in a sample to decay. The counter-intuitive part is that the proportion is fixed while the individual atom is entirely unpredictable: nothing about an atom indicates when it will go, there is no ageing and no build-up, and an atom that has survived a billion years is exactly as likely to decay in the next second as a fresh one.

Because the fraction is constant, the amount left falls by half each period regardless of how much you started with - so a sample never quite disappears, and dating works by measuring the ratio of parent to daughter atoms rather than the amount. Carbon-14, at 5,730 years, covers archaeology; potassium-argon, at 1.25 billion, covers the age of the Earth.

The concept has escaped into everything else. Drug half-lives determine dosing intervals; the biological half-life of caffeine is around five hours, which is why an afternoon coffee is still measurably present at midnight. It also turns up in economics and in the phrase about the half-life of a fact, describing how quickly published knowledge in a field becomes wrong - which is a metaphor, though a useful one, since unlike atoms facts do not decay at a constant rate.