I do believe that you are talking about filial imprinting, in which a young animal acquires several of its behavioral characteristics from its parent. It is most obvious in nidifugous birds, which imprint on their parents and then follow them around.
It was first reported in domestic chickens, by the 19th-century amateur biologist Douglas Spalding and was rediscovered by the early ethologist Oskar Heinroth, and studied extensively and popularised by his disciple Konrad Lorenz working with greylag geese.
Lorenz demonstrated how incubator-hatched geese would imprint on the first suitable moving stimulus they saw within what he called a "critical period" between 13–16 hours shortly after hatching. Most notably, the goslings would imprint on Lorenz himself (more specifically, on his wading boots), and he is often depicted being followed by a gaggle of geese who had imprinted on him.
Filial imprinting is not restricted to animals that are able to follow their parents, however; in child development the term is used to refer to the process by which a baby learns who its mother and father are. The process is recognised as beginning in the womb, when the unborn baby starts to recognise its parents' voices. (courtesy of Wikipedia
Filial Imprinting )
This occurs in a lot of animals, take for instance the cuckoo - it is brought up by birds of a different species as they are under the impression that it is their own chick
