Radioactive carbon used in carbon dating is carbon that is radioactive. Since it is active, it is unstable. Since it is unstable, it tends toward stability. This is common in nature; for example, if you hold an object at a certain height, it has the potential to fall, the potential energy that is. If you let go of it by simply removing your hand from underneath it, it would instantly become unstable; it would use up that potential to fall and fall toward the ground where it would be stable having no falling potential.
In the case of radiocactive carbon, its nucleus is not stable, and has a ground-like nucleus toward which it can “fall”. It therefore tends to transform to its ground-like stable version. Now, how does radioactive carbon, also called radiocarbon, enable us to date when was what was once a living organism alive?
Radioactive carbon combines with oxygen to form radioactive carbon dioxide which is then absorbed by plants leaves. Animals acquire the radioactive carbon thereafter by eating the plants. Now, radiocarbon, just like any radioactive element, has a half-life. Its half life is, if you take a certain quantity of radiocarbon, how long it will take for half the radiocarbon to transform to a more stable version.
Since it transforms to a different version, one might say that the initial version disappeared. Thus, after a half-life, half of of the initial radioactive carbon atoms would disappear. The half-life is therefore how long it takes half of the initial radioactive elements to disappear. The half-life of the radioactive carbon used in carbon dating, called carbon-14, is about 5730 years; meaning, if you take a million radioactive carbon-14 atoms and check back on them after 5730 years, you would only find half of them.
In all living matter, there are ten unstable radioactive carbon-14 atoms in every ten trillion carbon atoms. Suppose one finds the remains of what was once a living thing and examines one of its bones. When it was alive, it had ten radioactive carbon-14 atoms in every ten trillion carbon atoms. This was a constant ratio while it was alive because even though carbon-14 atoms disappear, there was an equal intake through plants eating. However, at the moment of death, there was no longer an intake, and the number of radioactive carbons-14 began to decrease.
Now, suppose one finds five unstable radioactive carbon-14 atoms in every ten trillion carbon atoms in the bone. That would imply that half of the radioactive carbon-14 disappeared; it must have taken 5730 years for that to happen. Thus, the remains must be of a subject that began losing radioactive carbon-14 5730 years ago, and hence died 5730 years ago. The remains must therefore be of a subject that was alive around 5730 years ago.