For any category $A$, the category of endofunctors $End(A)$ is monoidal with respect to the (horizontal) composition (the composition of functors and the Godement product for natural transformations).
Given a monoidal category $(C,\otimes,1,l,r,a)$ a (left or right) $C$-actegory is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on an author and context, the left coherent action of $C$ on $A$ is a morphism of monoidal categories $C\to End(A)$ in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict monoidal functor. Right coherent actions correspond to the monoidal functors into the category $End(A)$ with the opposite tensor product.
$C$-actegories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a strict 2-category $_C Act^c$. A monad in $_C Act^c$ amounts to a pair of a monad in $Cat$ and a distributive law between the monad and an action of $C$.
The notion of $C$-action (hence a $C$-actegory) is easily extendable to bicategories (see Baković‘s thesis).
Bodo Pareigis, Non-additive ring and module theory I. General theory of monoids, Publ. Math. Debrecen 24 (1977), 189–204. MR 56:8656; Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms, Publ. Math. Debrecen 24 (351–361) 1977.
M. Kelly, G. Janelidze, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61–91 link
P. Schauenburg, Actions of monoidal categories and generalized Hopf smash products, J. Algebra 270 (2003) 521–563 (remark: actegories with action in the strong monoidal sense)
Z. Škoda, Distributive laws for actions of monoidal categories, arxiv:0406310, Equivariant monads and equivariant lifts versus a 2-category of distributive laws, arxiv:0707.1609
If an actegory is like a module, then a biactegory is like a bimodule.