Although an element’s number of protons cannot change, the number of neutrons can vary slightly from each atom.
Most carbon on Earth exists as the very stable isotope carbon-12, with a very small amount as carbon-13. Carbon-14 is an unstable isotope of carbon that will eventually decay at a known rate to become carbon-12. But when gas exchange is stopped, be it in a particular part of the body like in deposits on bones and teeth, or when the entire organism dies, the ratio of carbon-14 to carbon-12 begins to decrease.

This plot shows the level of carbon-14 in the atmosphere as measured in New Zealand (red) and Austria (green), representing the Northern and Southern Hemispheres, respectively. Bomb radiocarbon dating is a term for radiocarbon dating based on timestamps left by above-ground nuclear explosions, and it is especially useful for putting an absolute age on organisms that lived through those events. The only major fluctuation [in carbon-14] we know of occurred when we began detonating nuclear weapons in the open air, back in the mid-20th Century. Most radiocarbon dating today is done using an accelerator mass spectrometer, an instrument that directly counts the numbers of carbon 14 and carbon12 in a sample.

A detailed description of radiocarbon dating is available at the Wikipedia radiocarbon dating web page.

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