The fibres of any morphism of schemes are schemes over a field. If the morphism if of finite type, then they are finite type schemes over a field, i.e. are (essentially) varieties.
Many arguments proceed from the case of a variety over a field, to the case of a scheme over an Artinian local ring, to a scheme over a complete local ring (a projective limit of Aritinian local rings), to a scheme over a local ring (say by descent), to the general case (by passing from stalks to neighbourhoods). For example, many results about abelian schemes are proved in this (or some closely related) way. (See e.g. Kevin Buzzard's answer.answer, and see also Brian Conrad's answer for an example of this kind of argument in a different context.)
