Carbon-14 is an isotope of carbon that is found naturally in our atmosphere. It is radioactive, and is best known because it collects in biological entities while they are living. Once the plant or animal dies, it stops taking in new carbon-14 and the carbon-14 it has starts decaying. By measuring the amount of carbon-14 left in the remains you can judge their age.
Carbon-14 is formed when radiation in the from of cosmic rays enters the atmosphere. Gamma radiation produces free neutrons, which fly about looking for a home. They often come to rest in a nitrogen atom, pushing out one of the protons and replacing it. Thus a typical nitrogen atom (seven protons and seven neutrons) becomes an atypical carbon atom (six protons and eight neutrons), giving it an atomic mass of 14, instead of the usual 12.
Although carbon-14 is produced primarily in the upper troposphere, it continuously disperses throughout the atmosphere. Along the way it joins with oxygen, as carbon will, to produce carbon dioxide. Plants take in this carbon dioxide, and the carbon is used in biological molecules of all sorts. Animals that eat these plants, and animals that eat the animals that eat these plants, pick up the carbon-14 and use it in their own bodies.
Once the plants or animals die, they stop taking in new building blocks (including carbon and carbon-14), and the carbon-14, being unstable, starts to decay. Carbon-14 in the atmosphere is also decaying continuously, of course, but this isn't so useful to us. Carbon-14 has a half-life of 5730 ± 40 years, so measuring the proportion of carbon-14 in a historical sample to the proportion of carbon-14 in a modern sample can give a good indication of the historical sample's age. Samples over the age of about 40,000 years have too little carbon-14 left to be of much use in dating them. Carbon-14 decays through beta decay (beta minus decay), in which one of the neutrons breaks down into a proton, an electron, and an anti-neutrino. The latter shoots off into the wild blue yonder, but the others stay and we once again have a typical nitrogen atom (nitrogen-14).
Carbon-14 can also be produced through other radiation related processes, both in the upper atmosphere and in laboratories, although these are minor sources. Recently the amount carbon-14 in the atmosphere spiked, as the radiation from nuclear testing in the 1940s, 50s, and 60s created excess amounts of carbon-14. This has the potential to mess up future carbon dating, but is otherwise harmless (there are some cases where this excess carbon-14 actually helps dating, as tooth enamel gives an exact record of how much carbon-14 was in the air at the specific time it was formed). Carbon dating is covered in its own node, but it is worth noting that the production of carbon-14 is not stable over time, and thus all carbon dating has some additional wiggle room on top of the statistical uncertainty of half-life predictions (the ± 40 years above). This has caused many archeologists and paleontologists major head aches, and is one of the reasons why scientists try to look at more than one method of dating.
Carbon-14 is, even after nuclear testing, only a very very very small part of our atmosphere, about 0.0000000001% of the carbon present on Earth.
Carbon-13 is a stable isotope of carbon, and is useless in carbon dating. It is apparently formed in the interior of stars, as is typical carbon, carbon-12. It has nothing to do with carbon-14, and was only included here because I was curious.