Read about the term isotope. An isotope is a specific form of an element. Each element's nucleus has two components: protons and neutrons. The amount of protons always remains the same, but the amount of neutrons can vary. For example, the most common isotope of carbon has six protons and six neutrons. However, the number of neutrons in a carbon atom can be more or less than six. One kind of isotope in carbon contains eight neutrons. This is called "Carbon 14." This isotope is not stable, which causes it to decay. The rate of decay in Carbon 14 and other unstable isotopes is called the "half-life."
Read about the term "half-life." The half-life of an isotope is always the same for each specific isotope. It is the amount of time it takes for half of the isotopes in an element to decay. For example the half-life for Carbon-14 is 5,730 years. So let's say that a certain sample of carbon started with 100 Carbon-14 isotopes. This sample will have 50 Carbon-14 isotopes 5,730 years later. There will be 25 Carbon-14 isotopes 5,730 years after that and so on. The rest of the atoms from the sample will become a more common form of carbon.
Measure the elemental composition using an isotope-ratio mass spectrometer (IRMS). The material you wish to date must contain an element with isotopes that have a half-life that is neither too long nor too short. For example, Oxygen-21 decays with a half-life of 3.42 seconds. If the material you are measuring is 100,000 years old, then the amount of Oxygen-21 will be too minute to make any kind of measurement. Uranium-238 has a half-life of 4.5 billion years, so it will have hardly changed at all in an artifact that was created in the last 5,000. In carbon, about one out of a every trillion atoms is a Carbon-14 isotope. As a point of reference, the thickness of a single strand of hair is over a million Carbon atoms. Each carbon isotope has a specific mass unto itself. Because of the extra two neutrons, carbon-14 is more massive than other carbon atoms. By measuring the total mass of an element with the IRMS, you can determine the ratio of a given isotope to other atoms.
Look at the ratio of isotopes in the material and compare that with the ratio that occurs naturally in the present. For example, all living things contain approximately the same percentage of Carbon-14. While the Carbon-14 is decaying it is being replaced by the food and air the plant or animal consumes. When the animal dies it stops replacing the isotope but it continues to decay at a regular rate. As a simple example, the same material as the material being dated but taken from a thing that is currently alive contains 100 carbon-14 isotopes. The material being dated contains six. This means that the material has gone through it's half-life about four times. If you take half of 100 four times you'll get 6.25. Since carbon-14 has a half-life of 5,730 years, you know the material being dated must be more than 22,920 (5,730 X 4) years old.