How does carbon-14 dating work? How can it tell a basket is 3000 years old, for example?

C14 is an isotope of carbon which is present naturally in all carbon either reacted or unreacted in very small amounts indeed (one atom in #10^12#. Whilst an organism is alive, carbon keeps entering and exiting (think carbon cycle) and so the quantities of C14 remains constant (this is one of the assumptions we make).

However, the second death occurs, the amount of carbon and therefore C14 in the sample becomes effectively 'trapped' and ceases to change. From that point on, the C-14 will effectively 'begin' its decay. By measuring the (radio) activity of a sample due to the C14 and comparing it to what it would have been at death, we can estimate the age of the sample. This works best i.e. is most accurate within time frames of a similar order of magnitude to the half life of the C14.

For example, a sample of organic material which shows an activity due to C14 which is one quarter of that of a current sample would have been around for two half lives and therefore will be estimated to be 11,400 years old.

NB half life is the time taken for half the radioactive atoms in a sample to decay. It is constant for any given isotope (remember radioactivity is totally random though).