In this study, a novel and physically realizable nanoscale communication paradigm is introduced based on a well-known phenomenon, Förster resonance energy transfer (FRET), for the first time in the literature. FRET is a nonradiative energy transfer process between fluorescent molecules based on the dipole-dipole interactions of molecules. Energy is transferred rapidly from a donor to an acceptor molecule in a close proximity such as 0 to 10 nm without radiation of a photon. Low dependence on the environmental factors, controllability of its parameters, and relatively wide transfer range make FRET a promising candidate to be used for a high-rate nanoscale communication channel. In this paper, the simplest form of the FRET-based molecular communication channel comprising a single transmitter-receiver nanomachine pair and an extended version of this channel with a relay nanomachine for long-range applications are modeled considering nanomachines as nanoscale electromechanical devices with some sensing, computing, and actuating capabilities. Furthermore, using the information theoretical approach, the capacities of these communication channels are investigated and the dependence of the capacity on some environmental and intrinsic parameters is analyzed. It is shown that the capacity can be increased by appropriately selecting the donor-acceptor pair, the medium, the intermolecular distance, and the orientation of the molecules.