The theory of recapitulation - otherwise known as the Biogenetic Law or Haeckel's law - is easily stated: in a phrase, "ontogeny recapitulates phylogeny".
In other words, it is the partly or wholly erroneous theory that an individual, during its development, passes through a series of stages corresponding to its successive evolutionary ancestors. While recapitulation is false as a theory, it is often observed in the development of an organism.
An individual develops by 'climbing up its family tree':
The transitory appearance of structures resembling gill slits in the development of humans, and other mammals, is a striking example. Mammals evolved from an ancestral fish stage and their embryonic gill slits recapitulate the piscine ancestry.
There are many examples of specialized larval forms that are not recapitulated ancestral stages - the caterpillar of Lepidoptera is an example. These exceptions notwithstanding, recapitulation is noticeably common.