If you know the rate that carbon-14 decays at, and how much of the carbon in a shroud, iceman or piece of old wood or bone is radioactive, you can work out how long ago they stopped breathing or photosynthesising. It just involves a bit of maths.
Of course, it helps if you have a valid sample and most radiocarbon dating experts who reference the Shroud of Turin know that the sample taken in 1988 is now known to have been invalid. See my comments in the right column.
Highly recommended: A date with carbon | The Archaeology News Network