Physicist: As far as carbon dating is concerned, the difference between living things and dead things is that living things eat and breathe and dead things are busy with other stuff, like sitting perfectly still. Eating and breathing is how fresh 14C (carbon-14) gets into the body.
The vast majority of carbon is 12C (carbon-12) which has 6 protons and 6 neutrons (12=6+6). 14C on the other hand has 6 protons and 8 neutrons (14=6+8). Chemically speaking, those 6 protons are far more important since they are what makes carbon act like carbon (and not oxygen or some other element). The extra pair of neutrons do two things: they make 14C heavier (by about 17%), and they make it mildly radioactive. If you have a 14C atom it has a 50% chance of decaying in the next 5730 years (regardless of how old it presently is). That 5730 year half-life is what allows science folk to figure out how old things are, but it’s also relatively short.
This begs the question: why is there any 14C left? There have been about 1,000,000 half-lives since the Earth first formed, which means that there should only be about of the original supply, which even google considers too small to be worth mentioning. The answer is that 14C is being continuously produced in the upper atmosphere.
Our atmosphere is brimming over with 14N. Nitrogen-14 has 7 protons and 7 neutrons, and is about four fifths of the air you’re breathing right now. In addition to all the other reasons for not hanging out at the edge of space, there’s a bunch of high-energy radiation (mostly from the Sun) flying around. Some of this radiation sometimes takes the form of free neutrons bouncing around, and when nitrogen-14 absorbs a neutron it sometimes turns into carbon-14 and a spare proton (“spare proton” = “hydrogen”).
This new 14C gets thoroughly mixed into the rest of the atmosphere pretty quickly, and carbon in the atmosphere overwhelmingly appears in the form of carbon dioxide. It’s here that the brand-new 14C enters the carbon cycle. Living things use carbon a lot (biochemistry is sometimes called “fun with carbon”) and this carbon enters the food chain through plants, which pull it from the air. Any living plant you’re likely to come across is mostly made of carbon (and water) it’s absorbed from the air in the last few years, and any living animal you come across is mostly made of plants (and other animals) that it’s eaten in the last few years.
With the notable exception of the undead, when things die they stop eating or otherwise absorbing carbon. As a result, the body of something that’s been dead for around 5700 years (the 14C half-life) will have about half as much 14C as the body of something that’s alive. Nothing to do with being alive per se, but a lot to do with eating stuff.
There are some difficulties with carbon dating. For example, nuclear tests or unusual solar weather can change the rate of production. Also, any attempt to measure things that have been dead for more than several half-lives (tens of thousands of years) are subject to a lot of statistical noise. So you can carbon date woolly mammoths, but definitely not dinosaurs. Aside from that, carbon dating is a decently accurate way of figuring out how long ago a thing recused itself from the carbon cycle.
Answer Gravy: This was subtle, and would have derailed the flow of the post, but extremely A-type readers may have noticed that adding a neutron to 14N (7 protons, 7 neutrons) leaves 15N (7 protons, 8 neutrons). But 15N is stable, and will not decay into 14C, or anything else. So why does the reaction “n+14N → p+14C” happen? It turns out that nuclear physics is more complicated than you might expect.
The introduced neutron can carry a fair amount of kinetic energy, and this extra energy can sometimes make the nucleus “splash”. It’s a little like pouring water into a glass. If you pour the water in slowly, then nothing spills out and the water-in-glass system is stable. But if you pour the same amount of water into the glass quickly, then some of it is liable to splash out. Similarly (maybe not that similarly), introducing a fast neutron to a nucleus can have a different result than introducing a slow neutron.
Dealing with complications like this is why physicists love themselves some big computers.