Glass-forming ability characterizes how easily we can avoid crystallization and thus make a glassy state of material from its liquid state upon cooling. Its quantitative measure is given by the critical cooling rate, which is the minimum cooling rate required for the formation of glass, or the avoidance of crystal nucleation. Here we consider what physical factors control the glass-forming ability of a liquid. We argue that short-range bond ordering, which is induced by a symmetry-selective part of the interaction potential, causes frustration against crystallization and helps vitrification if its local symmetry is not consistent with that of the equilibrium crystal. Based on this physical picture, we propose that the degree of short-range bond ordering in a liquid should be a new additional physical factor controlling the glass-forming ability. According to our model, it is also a controlling factor of the fragility, which characterizes how steeply viscosity increases upon cooling. This picture suggests a negative correlation between the glass-forming ability and the fragility. The close relationship between the glass-forming ability and the quasicrystal-forming ability in metallic glass formers is also discussed in the light of our model.