The statement "ontogeny recapitulates phylogeny" is credited to Ernst Haeckel, and was the credo and motivation for much embrological research in the 19th and early 20th century. Despite the fact that this is still taught to high school and university students, it has been thoroughly disproven and discredited. That is not to say, however, that the essential idea that an organism's evolutionary history is not reflected in its embryological development. In fact, the modern credo is instead "phylogeny recapitulates ontogeny", the reverse of Haeckel's idea and in fact the hypothesis originally put forth by Karl Ernst von Baer, Haeckel's predecessor.

The latter statement means, and what modern biologists believe, that the embryonic stages of an organism or species should resemble the embryonic (and not adult) stages of its ancestors. In other words, the embryonic development of any species is constrained to resemble (closely) the embryonic development of its ancestors. In other words, by examining the ontogeny of an individual species, we can infer certain aspects of their evolutionary history.

Consider the two following cases as evidence supporting the modern hypothesis. First, snakes and legless lizards develop 'leg buds' as embryos, only to have them re-absorbed prior to hatching. This same pattern is observed in whales and dolphins. In both cases, independent evidence shows that snakes evolved from legged ancestors, as did the whale and dolphin. A second example of the same type can be found in the development of the mammalian inner ear. In reptile embryos, two bones develop into the articular bones of the hinge of the jaw, while these two same bones become the hammer and anvil of the inner ear in marsupials. This suggests that, evolutionarily, the inner ear of mammals developed from bones originally found in the jaw of a common ancestor to the reptiles, and the fossil evidence clearly shows this to be the case.

Sources: many, many years of education as a biologist, supplemented with some information from the web site: