How is carbon-14 used in radiocarbon dating?
This is a very "cartoonish" explanation....because that is usually the way my brain works. Wikipedia will probably have some more details, but here's the idea.
When an organism is made (tree, mammoth, you and me, etc), an isotope of carbon, C-14, gets incorporated into the matter at about 1% compared to the other carbon, C-12. So in a mass of 100g of dry carbon, you'd have about 1g of Carbon-14.
Carbon-14 has a half-life of 5730 years, and it undergoes beta decay in its decay process (things are radioactive because of an unstable ratio of neutrons to protons...and Beta decay increases proton number by 1 and decreases neutron number by 1).
So in that 100g sample, if you held a Geiger counter up to it (thing that goes beep, beep in the presence of certain radioactive decay events), the sample would beep, beep, beep, beep.
If you came back 5730 years later, the same sample would beep.....beep (it would lose half of its beeps....or said another way, half of its radioactive C-14 are gone).
So you take a sample of carbon out of the ground and you weigh it. You find that it is 100g. It has been there a LONG time, and when you test if for radioactive C-14, you listen........beep.................................beep. This means that there is hardly any C-14 in it. You can use an equation to figure out how many half lives have passed (how much your beep, beep has diminished....because it should be 1% of sample), and then you multiply # of half lives by 5730 years and you've got a good idea how old that carbon sample is.