Elliptic curves can be defined over arbitrary base schemes $S$. Loosely speaking, what one gets is a family $E$ of elliptic curves parametrized by the points of $S$. One then proves the existence of the group law ($E$ can be given the structure of an $S$-group scheme), and goes from there.

There are indeed more than one ways to define this: one can for instance insist that all geometric fibers $E_{\overline{s}}$ are elliptic curves (this is done in Katz-Mazur, who define an elliptic curve as a proper smooth morphism $f : E \rightarrow S$ of finite presentation, with a section $0 : S \rightarrow E$, such that all geometric fibers of $f$ are integral curves of genus one). Alternatively, one may allow degenerate fibers (as in Deligne-Rapoport).