capture-mark-recapture is a statistical sampling method to estimate an unknown population size by taking two independent samples and gauging the degree of overlap between them.
A typical example of a capture-recapture method is as follows. Suppose we want to estimate the number of fish (of a given species, probably) in a lake. At some point in time, and at some location on the lake, we take a sample of fish (say $n_1=500$), mark them, and let them go. At a later point in time, and probably in a different location, we take another $n_2=500$ fish and count how many of them were in the original catch. If we found say $k_2 =20$ marked fish in the second sample, then our estimate of the total number of fish is $n_1 n_2 / k_2 = 12500$. More explicitly:
\begin{align} \frac{k_2}{n_2} &= \frac{n_1}{N} \\ \frac{20}{500} &= \frac{500}{N} \\ N &= \frac{500\times 500}{20} \\ N &= 12500 \end{align}
Capture-recapture wants both samples to be completely random because you expect that marked fish mingle with others in the closed volume. In other words, randomness of the two samples is comprised of (i) taking the samples of fish in randomly selected portions of the lake; (ii) relying on natural biological mixing processes to re-distribute the marked fish around. See Thompson (1992, Ch. 18).
Similar techniques have been used in other contexts, including analysis of networks (Warren, Airoldi and Banks 2008), hard to reach populations (Handcock, Gile and Mar 2013) and literary studies (Efron and Thisted 1976).
Efron, B. and Thisted, R. (1976). Estimating the Number of Unsen Species: How Many Words Did Shakespeare Know? Biometrika, Vol. 63, No. 3, pp. 435-447. http://www.jstor.org/stable/2335721
Handcock, M.S. Gile, K. and Mar, C. (2013) Estimating Hard-to-Reach Population Size Using Respondent-Driven Sampling Data. http://paa2013.princeton.edu/abstracts/130169
Thompson, S. K. (1992) Sampling. Wiley, New York.
Warren, R., Airoldi, E. and Banks, D. (2008). Network Analysis of Wikipedia. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.4966
