|Me, standing by some fresh water, in 2009|
There are two stable isotopes - carbon-12, with 6 neutrons, and carbon-13 with 7 neutrons. Carbon-12 is by far the most common isotope (thanks to it being made in stars in one of the key stellar nuclear fusion reactions), but carbon-13 still is around in appreciable amounts. It is interesting in its own right, and is used in all sorts of interesting ways but Carbon dating relies on an unstable isotope; Carbon-14.
Being unstable means that carbon-14 nuclei decay with an average half-life of about 6 000 years, meaning that if you have a sample of carbon-14, half of it will have decayed (to nitrogen) within one half-life. It is around on Earth in trace amounts, despite Earth being much older than 6 000 years, because it is formed in nuclear reactions in the the atmosphere. These reactions occur when cosmic rays (mostly protons) from space collide with atoms of atmospheric gas, producing neutrons, which then react with nitrogen-14, knocking out a proton, and causing carbon-14. The carbon reacts with oxygen to make carbon dioxide, which then makes its way in to the ecosystem by e.g. dissolving in water and falling as rain, which is then taken up by plants and animals. We each have a proportion of radioactive carbon-14 nuclei in our bodies as a result.
Once we die, though, we no longer take up new carbon-14. What carbon-14 is in our bones will then decay, and the proportion of carbon-14 to carbon-12 can be measured to tell how long it was since we (or whatever else we are measuring) was alive. This is the basis of carbon dating. It does rely on some assumptions - such as that the historic rate of carbon-14 production in the atmosphere has not changed much, that the resulting carbon-14 is equally distributed throughout the world, and that all living material takes up carbon-14 at the same rate as all other things. There are ways in which it is known that some of the assumptions are not so good, and corrections are made because of them. Various cross-checks can be made too, such as correlating carbon age with tree-ring age.
Thanks to a recent tweet by STFC's Head of Communications, I learned about a particular effect which can skew carbon dating results. The tweet pointed to this story on a science news website. The effect is to do with fish which have an anomalously low ratio of carbon-14 to carbon-12, which then get cooked by prehistoric people, leaving residue in the cooking vessels which can be carbon dated. Rather unhelpfully, the news story says "Hard water contains high levels of calcium, and calcium contains no Carbon-14." While clearly true (calcium is one element, carbon another), I'm not convinced about it as an explanation. There is a link on the page to the scientist behind the story, and while it seems most of the work has been presented at conferences with no write up, the abstract for one of the conferences describes the reasoning.
The effect is called the Freshwater Reservoir Effect. It happens because rivers can have large amounts of carbon-containing minerals dissolved in them (for example, if the source of the river flows over particular kinds of rock). These minerals are old, with no carbon-14. The proportion of carbon-14 in the rivers is then low, and the proportion in the fish living in the rivers also low. The proportion found when dating prehistoric clay bowls used to cook the fish in is also low. If not for the freshwater reservoir effect, one would conclude that the bowl was actually much older than it is, potentially by many hundred years.