Atoms floating around high in the Earth's atmosphere are continuously bombarded by energetic radiation (even the occasional cosmic ray). This high energy bombardment of the atmosphere knocks a few neutrons loose. These neutrons zip around randomly until they are captured by other atoms in the atmosphere. If they happen to be picked up by a nitrogen atom the reaction produces one atom of carbon-14 and one atom of hydrogen.

Most naturally occurring carbon has atomic weight 12. The nucleus is made of 6 protons and 6 neutrons and is very stable. About 1% of naturally occurring carbon contains an extra neutron and is called carbon-13. One in a trillion carbon atoms in the atmosphere is the carbon-14 described above. Given the large number of atoms in matter, there are still plenty of carbon-14 atoms floating around. Each 1 liter of the atmosphere contains about 9,000,000 carbon-14 atoms.

Carbon 13  (usually written as ¹³C in scientific papers) like ¹²C is stable. However ¹⁴C is not. It decays (via Beta decay) with a strict statistical probability to become ¹⁴N. In any given year each ¹⁴C atom has a 1 in 8,500 chance of decaying. That's like saying decay of a ¹⁴C atom is like an 8,500 year "storm."  It also means that in any given sample of carbon-14, it takes 5,730 years for half of the atoms to decay.

In 1949 physicist Willard Libby at the University of Chicago figured out that this phenomenon could be used to determine the age of various previously living things. He reasoned that as long as a plant or animal was alive, it was taking up carbon from the atmosphere including some ¹⁴C. When it died, the percentage of ¹⁴C in the organism matched the percentage in the atmosphere, but because the organism was no longer breathing and eating, any atoms that decayed were not replaced and the percentage began to drop. If you measure the percentage of ¹⁴C left in a sample and do the math, you will be able to determine how long ago the organism was alive. Libby worked out the details and was awarded the Nobel Prize in 1960.

Today we can determine when organisms that died up to 60,000 years ago lived. Beyond that there are uncertainties (particularly lack of knowledge of the amount of CO₂ in the atmosphere) that make this method unreliable. However, other dating techniques can be used to determine the age of older samples.