Why do currents, functionals on compactly supported differentiable n-forms, bear the name they do?
I've assumed that it has something to do with an electrical current being formalized as a vector field along a curve or other submanifold, but this is nothing but a guess.
George de Rham motivates the name "current" for his functionals by the fact that in three dimensions electric currents are represented by "1-dimensional currents".
The history of this concept is described by Jesper Lützen in De Rham’s Currents.
The classical electric current density can be modelled as a 2-form $$J=J_{ij}\wedge dx^{ij}$$ which is assumed to be locally integrable over a 3-manifold (3-dimensional domain) $X$. By integrating $J$ over a 2-submanifold (a surface) $S\subset X$, one gets the electric current through S, i.e. $$I(S)=\int_{S}J,$$ that is a simple example of a 2-current in the sense of de Rham. Extending this notion to the $n$-dimensional case, one can naturally model a density on the $n$-manifold as a twisted $(n − 1)$-form $J\in\Omega^{n-1}(X,L)$, and to treat the corresponding integral $I(S)$ as a generalized electric current through the (n-1)-dimensional submanifold $S\subset X$.
Whether this was a real motivation behind the notion of de Rham's currents, I don't know.
I am not sure if this helps, but elaborating on Andrey's answer: now if you recall that occasionally one wants to work with electric currents localised on submanifolds of $\mathbb{R}^3$, what a physicist would call a "wire", or a "conducting plane", de Rham currents provide a natural framework for this. This is much like using distributions to describe point charges.
For example, a current corresponding to a loop wire carrying electric current $I$ represented by an oriented curve $\gamma$ is the 1-current $J$ given by $J(\phi)=I\int_\gamma\phi$. This can be though of as a "generalized" 2-form $J$, which we now can try to integrate over 2-manifold $S$ to obtain the current through this surface, but I am not sure that it makes rigourous sence for arbitrary 1-currents (for this particular one it does, at least for finite number of transversal intersections of $\gamma$ and $S$). We can also take $J(E)=RI^2$, where R is the resistance, and $E$ is the electric field 1-form to express the Ohm's law, et cetera.
Because they run: Correr, corriente, in spanish is the same word for running water, corrientes de agua, corrientes eléctricas ...
