Let $G$ and $H$ be two groups. We say $G$ acts on $H$ if there is a group homomorphism $\rho:G\longrightarrow \textrm{Aut}(H)$, where the $\textrm{Aut}(G)$ is the group of automorphisms of $H$.
We can regard $G$ and $H$ as the sets of morphisms of small categories with only one object. The composition and identities are induced from the group definition. So we can say an action of $G$ on $H$ is an action of the category $G$ on the category $H$.
In general, if $\mathcal{C}$ and $\mathcal{D}$ are categories, is there a notion of an action of $\mathcal{C}$ on $\mathcal{D}$?
Thanks.
In Introduction to foliations and Lie groupoids (§5.3), Moerdijk and Mrcun define the action of a groupoid on another groupoid, which is already a generalization of the situation you describe.
A (right) action of a groupoid $G$ on a groupoid $H$ is given by two (right) actions of $G$ on $H_1$ (the set of morphisms of $H$) and on $H_0$ (the set of objects of $H$) such that the groupoid structure maps of $H$ are equivariant.
Moreover, Julia Bergner has developped a notion (a rooted action) of a category acting on another category in her article Reedy categories which encode the notion of category actions.
She defines the action of a small category $\mathcal{C}$ on a set $A$, and then a rooted action of a category $\mathcal{C}$ on a category $\mathcal{D}$ is given by an action of $\mathcal{C}$ on the set of morphisms of $\mathcal{D}$, satisfying two additional axioms.
However, it seems that the latter is not a generalization of the former, since if $\mathcal{D}$ is a discrete category then there is no nontrivial rooted action of $\mathcal{C}$ on $\mathcal{D}$, though there may be nontrivial actions of $\mathcal{C}$ on $\mathcal{D}_0$.