The electronic and magnetic properties of Fe on Ir(111) have been investigated experimentally by spin-polarized scanning tunneling microscopy (SP-STM) and theoretically by first-principles calculations based on density functional theory. While the growth of an Fe monolayer is in-plane commensurate, deposition of a double-layer shows a rearrangement of atoms due to strain relief accompanied by local variations of the electronic structure. Both stackings of the monolayer, i.e. face centered cubic (fcc) and hexagonal closed packed (hcp), are observed experimentally. The magnetic structure of both types is imaged with SP-STM. From these experiments, we propose a nanoscale magnetic mosaic structure for the fcc-stacking with 15 atoms in the unit cell. For hcp-stacking, the tunneling spectra are similar to the fcc case, however, the magnetic contrast in the SP-STM images is not as obvious. In our first-principles calculations, a collinear antiferromagnetic (AFM) state with a 15 atom in-plane unit cell (AFM 7 : 8 state) is found to be more favorable than the ferromagnetic state for both fcc- and hcp-stacking. Calculated SP-STM images and spectra are also in good agreement with the experimental data for the fcc case. We performed spin spiral calculations which are mapped to a classical Heisenberg model to obtain the exchange-interaction constants. From these calculations, it is found that the AFM 7 : 8 state is energetically more favorable than all solutions of the classical Heisenberg model. While the obtained magnetic exchange constants are rather similar for the fcc and hcp stacking, a comparison with the experiments indicates that competing interactions could be responsible for the differences observed in the magnetically sensitive measurements.