Embryology is the study of embryos and their development. The study of embryology starts with the fertilisation of an egg, and continues until the foetus stage. A broader term, developmental biology, covers the whole period of growth from the egg to adult life.
In mammals (and some other animals) embryos develop inside the mother, in the mother's womb. Otherwise it takes place after the egg is laid. It most animals, there are various stages between the egg and the adult animal. If the stages are well-defined the process is called metamorphosis.