Determining the age of artifacts has helped humans to learn more about history, and about the civilizations and ages that came before the present day. For example, if you take the example of an ancient artifact (such as a fossil), it's age can be used to pinpoint to which civilization it belonged (or the inverse if the civilization it is from is known, and scientists are looking for the time period of that civilization). To achieve this, scientists use a method called carbon dating.
This method is based on the fact that all things that once lived contain carbon. Usually, carbon is in a stable form of 12-carbon, but sometimes it can occur in a radioactive isotope called 14-carbon. Since this isotope is radioactive, it will in time undergo radioactive decay. Carbon dating uses this to pinpoint the age of an artifact. It works by calculating the decay rate of the 14-carbon in the artifact. Our knowledge of nuclear decay tells us that since decay is random and spontaneous, (i.e. not affected by outside factors), the rate of decay will decrease over time.
Using nuclear decay formulae, scientists can calculate how long the artifact has been undergoing radioactive decay using the ratio of 14-carbon nuclei in the material (assumed to be the same as atmospheric ratio), and the known value of it's half life (the time taken for half of the sample to decay). The result is known as the raw BP age, or the age Before Present (where present is taken to be around 1950 to keep consistent) and is measured in raw radiocarbon years.
However, the result is not always completely accurate. Calibration is needed to factor in changes such as in cosmic ray intensity, climate changes, or human activities. Also, there are large reservoirs of carbon (both in 12- and 14- form) in organic matter, such as ocean sediments, which could affect changes in the atmospheric ratio and affect the results. By including these, the formula can be calibrated and improve the accuracy of the result, giving what is referred to as the calibrated date. It is these calibrated dates that scientists use as calendar dates when determining the age of artifacts.
However, carbon dating is not perfect. The main reason for this is the necessity of the material to have once-lived. While this will include most tools (using wood in some way) or fossils, it cannot date non-organic materials, due to the lack of carbon that is needed to be able to decay. Also, when carbon dating some artifacts, such as vases, the method can only be accurate to when the clay used in the making of the vase was formed, and not when the actual clay was fired and shaped.
Despite this, carbon dating is still used in science today to help us get a better understanding of the history of our world.