The stable isotope of carbon, carbon-12, has 6 protons and 6 neutrons (adding to 12). Carbon-14 has two extra neutrons, and is unstable. Carbon-14 is produced at a fairly constant rate by the interactions of cosmic rays with the upper atmosphere, so while there are only trace amounts of #""^ 14C# in the amosphere (as #CO_2#), the amount seems to be stable over time.
As plants photosynthesize, they incorporate #""^ 14C# into their cells, and animals that eat those plants then also get #""^ 14C#.
Because of constant turnover of atoms in living things, the amount of #""^ 14C# in living organisms is about the same as in the atmosphere. But once the organism dies, no fresh #""^ 14C# is incorporated, and whatever is there will slower disappear through radioactive decay.
#""^ 14C# has a half life of about 5730 years, meaning that if you have 1g of #""^ 14C#, after 5730 years half of it would be gone, leaving you only 0.5g of #""^ 14C#.
Thus by measuring the concentration of #""^ 14C# in a sample of organic matter, scientists can tell by how much remains how long ago the organism died. This works for things like cotton and wood as well as dead tissue, but does not work on rocks.
Because the amounts of #""^ 14C# are fairly small, after about 60,000 years there is no measurable #""^ 14C# left, so it is only useful for analysis of relatively recent samples, geologically speaking. It can't be used to get the age of dinosaurs.