A cochain complex $V^\bullet$ in an Ab-enriched category $C$ is nothing but a chain complex in $C$, but with the sign of the $\mathbb{Z}$-grading reversed.
This means that for a cochain complex $(V^\bullet,d)$ the differential (or coboundary) $d$ increases the degree
An archetypical example is the deRham cochain complex of differential forms on a manifold $X$, or more generally of differential forms in synthetic differential geometry.
The dual (in some suitable sense) of a chain complex $(V_\bullet, \partial_k : V_k \to V_{k-1})$ is canonically a cochain complex with $V^k := (V_k)^*$ and $d_k := (\partial_k)^*$.
A monoid in chain complexes as well as cochain complexes is a differential graded algebra.
By the dual Dold-Kan correspondence cochain complexes in non-negative degree are equivalent to cosimplicial abelian groups.