Carbon has two stable isotopes: Carbon-12 (12C), and Carbon-13 (13C). In addition, there are tiny amounts of the unstable isotope Carbon-14 on earth. Carbon-14 has a half life of just under 6000 years, and so would have long ago vanished from the earth, were it not for its constant formation by cosmic ray impacts on nitrogen in the atmosphere. Plants take up atmospheric carbon dioxide by photosynthesis, and are eaten by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radiocative decays. This decay can be used to get a measure of how long ago a piece of once-living material died.
Measurements were originally made by counting the radioactive decay of individual carbon atoms, but this was relatively insensitive and subject to statistical errors: there is never much carbon-14 to begin with, and a half-life that long means that very few of the atoms will decay while you're trying to detect them. Sensitivity and accuracy have since been greatly increased by the use of mass-spectrometric techniques, where the carbon-14 atoms can be counted directly.
Raw radiocarbon measurements are usually reported as years "before present" (BP). This is the number of "radiocarbon years" before 1950, based on the assumption that the level of carbon-14 in the atmosphere has always been at the 1950 level. The value will be subject to measurement uncertainty, e.g., 3000+-30BP indicates a standard deviation of 30 radiocarbon years.
The raw BP date can not be used directly as a calendar date, because the assumption that the level of carbon-14 remains constant does not hold true in practice. The level is maintained by high energy particles interacting with the earth's upper atmosphere, which may be affected by changes in the earth's magnetic field or in the cosmic ray background. It has also been affected by human activities - it was almost doubled for a short period due to atomic bomb tests in the 1950s and 1960s and has been reduced by release of CO2 from ancient organic sources with little carbon-14.
The BP dates are therefore calibrated to give calendar dates. Standard calibration curves are available, based on comparison of radiocarbon dates with other methods such as examination of tree growth rings and ice core samples. The difference between the Julian calendar and the Gregorian calendar can apparently be ignored, because it's insignificant compared to the measurement uncertainty.
Libby vs Cambridge half-life
Carbon dating was developed by a team led by Willard Libby. Originally a carbon-14 half-life of 5568+-30 years was used, which is now known as the Libby half-life. Latter a more accurate figure of 5730+-40 years was measured, which is known as the Cambridge half-life. However laboratories continue to use the Libby figure to avoid confusion. An uncallibrated dating using the Libby figure could be improved by multiplying by the ratio of these numbers (approximately 1.03), but this is usually unnecessary since the adjustment is included in modern calibration curves.
Examples of Carbon Dating and Historical Disputes
Carbon dating was used to determine the age of the material in the Turin shroud, and found an age approximating to the Middle Ages, around the time of the first known accounts of the shroud. making it around a thousand years newer than its supposed origin. However, some proponents of the shroud, ignoring Occam's razor have advanced the theory that the Resurrection caused the release of a large number of neutrons, that generated excess 14C in the cloth.
The website of the Oxford Radiocarbon Accelerator Unit has further details: http://www.rlaha.ox.ac.uk/orau/.