A question about current and Dirac measure $0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable? 
 A: Let $\mathcal{D}^p(X)$ denote the space of compactly-supported smooth $p$-forms on a smooth manifold $X$ (topologized in the same way as the space of distributions), then a $p$-current is defined to be a continuous linear functional $\mathcal{D}^p(X) \to \mathbb{R}$. 
An important class of currents arise from the oriented $p$-submanifolds $S \subset X$ (i.e. objects over which a $p$-form can be integrated): the current $[S]$ associated to the $p$-submanifold $S$ is defined to be the functional
$$
\eta \mapsto [S](\eta):=\int_S \eta.
$$
Furthermore, the $0$-currents are just linear functionals $\mathcal{D}^0(X) = \mathcal{C}^{\infty}_c (X) \to \mathbb{R}$, i.e. distributions on $X$. Those $0$-currents that arise from $0$-submanifolds can be easily described; for example, the current associated to the point $\{ P \} \subset X$ is given by 
$$[P](\eta) = \eta(P), \hspace{2mm} \eta \in \mathcal{D}^0(X) = C^{\infty}_{c}(X),
$$ as integration over a point just picks out the value of the integrand at that point. That is, the $0$-current $[P]$ is precisely the Dirac delta distribution on $X$ at the point $P$.
Edit: it is interesting to note how Stokes theorem behaves in this context: it says that for a $p$-dimensional oriented submanifold $S \subset X$ and $\eta \in \mathcal{D}^{p-1}(X)$,
$$
[S](d\eta) = \int_S d\eta = \int_{\partial S} \eta = [\partial S](\eta).
$$
This has a continuous extension to an operator $\partial \colon [\mathcal{D}^{p}(X)]' \to [\mathcal{D}^{p-1}(X)]'$ on currents, which is often called the boundary operator.
