We investigate the microscopic nature of the magnetism in  honeycomb iridium-based systems. We show that the minimal model describing  the magnetism in A$_2$IrO$_3$ includes both isotropic and anisotropic Kitaev-type spin-exchange interactions between nearest and next-nearest neighbor  Ir ions, and that the magnitude of the Kitaev interaction between next-nearest neighbor Ir  magnetic moments is comparable with nearest neighbor interactions. We computed  the low temperature phase diagram of the  effective model  with  classical Monte Carlo simulations. Due to the presence of the  anisotropic Kitaev interactions and the frustration introduced by the competition of the spin couplings between nearest and next-nearest  neighbors,  the  resulting phase diagram  is very rich. It contains both various commensurate states  and incommensurate  single-Q and multi-Q phases, whose  regions of stability are controlled by the ratios between competing exchange constants.  We showed that the second neighbor Kitaev term plays an important role in the stabilization of the  commensurate antiferromagnetic zigzag phase which has been experimentally observed in Na$_2$IrO$_3$. In our simulations, we found this phase to be the ground state for parameters of the model of
both the correct signs and magnitudes.