The film growth of Cu on clean and O-precovered Ru(0001) substrates at temperatures between 300 and 450 K is studied by means of scanning tunneling microscopy. On clean Ru(0001), the Cu films grow in a multilayer mode. For an O precoverage, (theta(0)) < 0.1 monolayer (ML), O remains trapped at the Cu/Ru interface and the Cu film grows similarly as on clean Ru(0001). Precovering the Ru(0001) substrate with more than 0.1 ML of O strongly modifies the film morphology. The excess O migrates to the surface of the growing film and acts as a surfactant. Domains of an O/Cu structure are formed, the lateral extension of which linearly increases with theta(0). For 0.4 < theta(O) less than or equal to 0.5 ML, the O/Cu structure covers the film surface completely. For 0.2 less than or equal to theta(0) less than or equal to 0.5 ML, a perfect layer-by-layer growth with a relatively high nucleation density is forced at temperatures around 400 K. Decreasing the temperature and/or theta(0) yields multilayer growth. For 0.4 < theta(O) less than or equal to 0.5 ML, temperatures above 430 K, and substrate terrace widths below 100 nm, step-flow growth is observed. Two different types of O/Cu surfactant structures (A- and B-type) are identified. The A-type structure is established for 0.1 < theta(0) less than or equal to 0.4 ML, and displays some ordering on a local scale (distorted hexagonal lattice). It causes heterogeneous nucleation at surface sites formed by a misfit-induced moire-like relaxation of the Cu film. Its surfactant effect can be described by the concept of two mobilities, which is based on a low adatom mobility during nucleation, and a high adatom mobility on top of small islands. This implies an increase of the attempt rate of Cu adatoms for step descent, enhancing interlayer diffusion. The B-type structure is established for 0.4 < theta(0) less than or equal to 0.5 ML, and contains a more irregular arrangement of O atoms. We assume that it behaves like a continuous O/Cu layer, on top of which the adatoms migrate. Its surfactant effect is attributed to a large diffusion barrier and a small additional step-edge barrier prevailing on top of the O/Cu layer. The latter also implies increased interlayer diffusion. The relationship of our results to previously observed work-function oscillations in the O-modified Cu film growth on Ru(0001) is discussed.