In formal logic and model theory, interpretation refers to equipping the syntax of some theory with a semantics.
Specifically in the context of categorical logic given a theory $T$ then an interpretation for it in some category $\mathcal{C}$ is a functor $Syn(T) \longrightarrow \mathcal{C}$ from the syntactic category of $T$ to $\mathcal{C}$ and preserving some relevant structure, such as limits. (See also at algebraic theory and algebra over an algebraic theory for more on this.
formula?
wikipedia interpretation (model theory)
Wilfrid Hodges, A shorter model theory, Cambridge Univ. Press 1997