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Radioactive dating (or radiometric dating) is determining a past date using the rate of radioactive decay of one or more of a material's components. The rate of decay of various nuclides is well known, so ratios of parent nuclides and daughter nuclides (the decay's decaying nuclide and the resulting nuclide) over time are clear. If the ratio at the time of some past event can be deduced, the time interval from that event to the present can be calculated.
Carbon dating uses the ratio of two isotopes of carbon: the radioactive isotope (carbon-14) is constantly generated by cosmic rays striking the upper atmosphere. It is incorporated into living beings throughout their life, but after death, the amount of carbon-14 in the organism decays, continually falling, directly affecting the ratio within the organism. The ratio in the atmosphere is roughly constant (with some recent changes due to modern technology) and its history going back thousands of years has been established through analysis of the carbon isotope ratios found in tree rings.
Rocks generally form as temperatures drop to the point that the material solidifies. The initial ratio of the daughter isotope (i.e., the result of radioactive decay) with other isotopes of the same element at the time of the cooling can be deduced by comparing multiple compounds in a rock that each has a different ratio of the elements. A pair of such compounds with different ratios is sufficient to pinpoint the age, and three or more offer independent checks. Compounds that solidify at different temperatures can offer information on multiple heating/cooling events, i.e., additional history.
Carbon dating is used to date objects going back thousands of years, often man-made objects. Dating minerals uses slowly-decaying materials that offer dating to billions of years, and the dating of various material including lunar rocks, meteorites, and Earth rocks consistently point to an age of the Earth and solar system as in the range of 4 to 5 billion years, with 4.568 billion years a cited number.