In this study, we introduce BEORN (Bubbles during the Epoch of Reionization Numerical Simulator), a publicly available PYTHON code that generates three-dimensional maps of the 21-cm signal from the cosmic dawn and the epoch of reionization. Built upon N-body simulation outputs, BEORN populates haloes with stars and galaxies based on a flexible source model. It then computes the evolution of Ly α coupling, temperature, and ionization profiles as a function of source properties, and paints these profiles around each source on to a three-dimensional grid. The code consistently deals with the overlap of ionized bubbles by redistributing photons around the bubble boundaries, thereby ensuring photon conservation. It accounts for the redshifting of photons and the source look-back effect for the temperature and Ly α coupling profiles that extend far into the intergalactic medium to scales of order 100 cMpc. We provide a detailed description of the code and compare it to results from the literature. After validation, we run three different benchmark models based on a cosmological N-body simulation. All three models agree with current observations from ultravoilet luminosity functions and estimates of the mean ionization fraction. Due to different assumptions regarding the small-mass stellar-to-halo relation, the X-ray flux emission, and the ionizing photon escape fraction, the models produce unique signatures ranging from a cold reionization with deep absorption trough to an emission-dominated 21-cm signal, broadly encompassing the current uncertainties at cosmic dawn. The code BEORN is publicly available at https://github.com/cosmic-reionization/BEoRN.