Where a simplicial object is a functor $\Delta^{op} \to \mathcal{C}$ out of the opposite category of the simplex category, a cosimplicial object is a functor $\Delta \to \mathcal{C}$ out of the simplex category itself.
When $\mathcal{C}$ has finite limits and finite colimits, then $\mathcal{C}^{\Delta}$ is canonically a simplicially enriched category with is tensored and powered over sSet. This is called the external simplicial structure in (Quillen 67, II.1.7). Review includes (Bousfield 03, section 2.10).
More generally, for any $\mathcal{C}$, we can make $\mathcal{C}^{\Delta}$ into a simplicially enriched category using the end formula
with composition inherited from $\mathcal{C}$ and $\Delta$.
Daniel Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967
Aldridge Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories (arXiv:0312531)