Molecular dynamics (MD) simulation has been used extensively to study water surfaces. Nevertheless, the quantitative prediction of water surface tension has been controversial, since results from different simulation studies using the same water model may differ considerably. Recent research has suggested that bond flexibility, long-range electrostatic interactions, and certain simulation parameters, such as Lennard-Jones (LJ) cutoff distance and simulation time, may play an important role in determining the simulated surface tension. To gain better insight on the MD simulation of water surfaces, particularly on the prediction of surface tension, we examined seven flexible water models using a consistent set of simulation parameters. The surface tensions of the flexible, extended simple point charge (SPCE-F) model and the flexible three-center (F3C) model at 300 K were found to be 70.2 and 65.3 mN/m, respectively, in reasonable agreement with the experimental value of 71.7 mN/m. More importantly, however, detailed analysis of the interfacial structure and contributions from various interactions have revealed that the surface tension of water is determined by the delicate balance between intramolecular (bond stretching) and intermolecular (LJ) interactions, which reflects both the molecular orientation in the interfacial region and the density variation across the Gibbs dividing surface (GDS). In addition, the water molecules on the liquid side of the GDS were found to lie almost parallel to the surface, which helps to clarify the dual-layer structure suggested by sum-frequency generation spectroscopy. By correlating the simulated surface tensions of the seven water models with selected molecular parameters, it was found that the partial charge distribution in the water molecule is likely a key factor in determining the near-parallel alignment of water molecules with the surface.