  A monolayer of molecules or quantum dots sandwiched between electrodes can be
driven out of equilibrium by the application of a bias voltage between the
electrodes. We study charge ordering, i.e., the spontaneous formation of a
charge density wave, and the perpendicular current in such a system within a
master-equation approach augmented by mean-field and classical Monte Carlo
methods. Our approach is suitable for weak tunneling between the monolayer and
the electrodes. For a square lattice with nearest-neighbor Coulomb repulsion,
we present a comprehensive study of the zero-temperature phases controlled by
the on-site energy, the bias voltage, and the degeneracy of the occupied
single-site state. One of the most interesting results is the prediction of a
conducting charge-density-wave phase that only occurs at a finite bias voltage.
We also study the universality classes of the phase transitions towards
charge-ordered states at zero and nonzero temperatures. While all transitions
at $T>0$ and some at $T=0$ belong to the two-dimensional Ising universality
class, we also find an absorbing-to-active phase transition in the
$\mathbb{Z}_2$ symmetric directed percolation (DP2) class at $T=0$.
