A rare-earth compound on a metal may form a two-dimensional (2D) intermetallic compound whose properties can be further modulated by the underlying substrate periodicity and coupling. Here, we present a combinational and systematic investigation using scanning tunneling microscopy/spectroscopy (STM/STS) and density functional theory (DFT) calculations on erbium (Er) on Cu(111). Experimentally, an intriguing growth mode transition from a branched island to a fractal-like island has been observed depending on whether the deposition process of Er is interrupted for a certain duration: post-deposition effects, such as nucleation and island growth controlled by diffusion, play an essential role in altering the Er island edge and its activity. Upon annealing, the branched Er islands become strands of amorphous surface alloy; in contrast, the fractal-like islands (with additional Er atoms on top) give rise to a monolayer thick 2D ErCu2 intermetallic compound and display a moiré pattern. Theoretically, using DFT calculations, we found that the characteristic energy states, particularly the state in the unoccupied region around 582-663 meV, of the 2D ErCu2 intermetallic compound are position-dependent, consistent with STS measurements. The moiré pattern originating from the mismatch of the periodicities of the ErCu2 layer and the Cu(111) surface was identified to be responsible for the observed periodic modulation on the coupling interaction that affects the electronic structures. Our further DFT calculations on a free-standing ErCu2 monolayer found it to be a 2D ferromagnet with topological band structures. Our work should stimulate further studies on such 2D rare-earth-based nanostructures and exploration of the use of the tunable electronic structures in such atomically-thin layers.