One typically starts with a monoidal category$M = (M, \otimes, 1)$ which is regular in the sense that it has equalizers which are preserved by the tensor product$\otimes$ on both sides. (The monoidal structure does not need be symmetric.)

There is a bicategory$Comod(M)$ of comonoids and bicomodules in $M$: this is a special case of the bicategory $Mod(K)$ of monads and bimodules in a bicategory $K$, where in this case $K = \mathbf{B} M^{op} = (\mathbf{B} M)^{co}$. Then an internal category in $M$ is a monad in $Comod(M)$.

Because every set is canonically a comonoid with respect to the cartesian product, a comonoid in Set is just a set and a bicomodule is a span, and a monad in the bicategory of spans of sets is just a small category. More generally, an internal category in the above sense in any category with finite cartesian products (and equalizers, of course, hence a finitely complete category) is just an internal category in the usual sense.