In the context of approximating second order boundary value problems we invented the name "de Rham map" with A. Bossavit in paper T. Tarhasaari, L. Kettunen, A. Bossavit: Some realizations of a discrete Hodge operator: A reinterpretation of finite element techniques. IEEE Trans. Magn., Vol 35(3), 1999, pp. 1494-97.
Intuitively, the idea is as follows. Finite element and finite difference kind of methods provide one with approximative solutions of boundary value problems in finite dimensional spaces. To build such approximations one needs a map from the (infinite dimensional) linear spaces of fields to some finite dimensional spaces in which the fields are approximated.
Having a "mesh", that is, a cellular complex, and a field, say p-form $f$, the de Rham map sends $f$ to integrals of $f$ on p-chains (which in turn are formal sums of p-cells of the complex/mesh). Or, alternatively, a simplified view, the de Rham map sends f to an array of integrals of $f$ on the (oriented) p-cells of the complex.
For example, the de Rham map sends the 2-form magnetic flux $b$ to an array of real numbers representing fluxes $\int_c b$ on the 2-cells $c$ (such as triangles) of the complex/mesh. In terms of classical language, if magnetic flux density ${\bf B}$ is considered as a (smooth) vector field, pair $({\bf B}, c)$ is mapped to $\int_c {\bf B}\cdot {\bf n}\, {\rm da}$.
Formally, the definition is something like: Let $F^p$ be the space of differential forms on manifold $\Omega$, $C^p$ the space of $p$-cochains in cellular complex $K$, and $K$ is a cellular tessellation of $\Omega$. Then, map $\mathcal{C}$ from $F^p$ to $C^p$ is the de Rham map, if for all $p$-chains $c\in K$ and smooth $p$-forms $f\in F^p$ map $\mathcal{C}$ satisfies $$\mathcal{C}f: C^p \rightarrow \mathbb{R}, \quad c\mapsto \int\limits_c f\,.$$
