The definition can be found in for example Section 3.3 of T. Bloom, M. Herrera: De Rham Cohomology of an Analytic Space (Inventiones math. 7, 275-296 (1969)) and Section 4.2 of M. Herrera, D. Lieberman: Residues and Principal Values on Complex Spaces (Math. Ann. 194, 259--294 (1971)) (in the more general setting of semianalytic sets).
If $M \subseteq U$, where $M$ is an analytic subset of an open set $U \subseteq \mathbb{C}^n$, and $\mathcal{E}_U$ denotes the sheaf of smooth forms on $U$, then one may first define $N_{M,U} \subseteq \mathcal{E}_U$ as the sheaf of smooth forms on $U$ whose pullback to the regular part of $M$ vanishes. Then one defines the sheaf of smooth forms on $M$ as $\mathcal{E}_M := \mathcal{E}_U/N_{M,U}$. This construction then globalizes to define smooth forms on an analytic space $\mathcal{E}_X$ which on a local model $M \subseteq U$ coincides with $\mathcal{E}_M$ defined above.
Then, one equips the space $\Gamma_c(U,\mathcal{E}_U)$ of sections with compact support with the usual appropriate (locally convex inductive) topology induced by uniform convergence of the coefficients of the form and of their derivatives. Since the kernel of the projection $\Gamma_c(U,\mathcal{E}_U) \to \Gamma_c(M,\mathcal{E}_M)$ is closed, one may equip $\Gamma_c(M,\mathcal{E}_M)$ with the quotient Hausdorff topology. Finally, for a general analytic space $X$, one equips $\Gamma_c(X,\mathcal{E}_X)$ with the appropriate topology which induces the given topology on the local models.
Regarding the question of what properties these currents have, this seems to be way broad question to answer here, but would have to be checked on a case by case basis.
One word of caution is that the definition in Bloom-Herrera is actually stated in two ways, claimed to be equivalent to each other, where the other definition that $N_{M,U}$ is alternatively defined as the space of smooth forms $\alpha$ on $U$ such that for any smooth map $g : W \to U$, with $W$ a manifold and $g(W) \subseteq M$, $g^*\alpha = 0$. As far as I know, the equivalence of these two definitions is only adressed recently in M. Andersson, H. Samuelsson Kalm: A note on smooth forms on analytic spaces (Math. Scand. 127 (2021), 521–526). With this alternative definition, it is straight-forward that a smooth map $f : N \to M$ of analytic spaces induces a pullback map $f^*$ of smooth forms, and thus a push-forward of smooth proper maps.