Carbon-14 is produced in the upper atmosphere when cosmic rays bombard nitrogen atoms. Plants take up C-14 along with other carbon isotopes during photosynthesis, and animals acquire C-14 by eating the plants (or other animals). During the lifetime of an organism, the amount of C-14 in the tissues remains at an equilibrium. However, when the organism dies, the amount of C-14 declines at a rate known as its half-life. Libby calculated the half-life of C-14 as 5568 ± 30 years. This means that half of the C-14 has decayed by the time an organism has been dead for 5568 years, and half of the remainder has decayed by 11,136 years after death. Radiocarbon dating depends on measuring the amount of C-14 in a sample and then calculating the age since death by considering the half-life.

Subsequent work has shown that the half-life of radiocarbon is actually 5730 ± 40 years, a difference of 3% compared to the Libby half-life. However, to avoid confusion all radiocarbon laboratories continue to use the half-life calculated by Libby, sometimes rounding it to 5570 years.

Further reading:

Bowman, Sheridan.
1990 Radiocarbon dating. London: British Museum Publications.

Taylor, R.E.
1987 Radiocarbon dating: an archaeological perspective. Orlando: Academic Press.

http://c14.sci.waikato.ac.nz/webinfo/