The horizontal-branch (HB) morphology of globular clusters (GCs) is mainly determined by metallicity. However, the fact that GCs with almost the same metallicity exhibit different HB morphologies demonstrates that at least one more parameter is needed to explain the HB morphology. It has been suggested that one of these should be a global parameter that varies from GC to GC and the other a nonglobal parameter that varies within the GC. In this study we provide empirical evidence corroborating this idea. We used the photometric catalogs obtained with the Advanced Camera for Surveys of the Hubble Space Telescope and analyze the color-magnitude diagrams of 74 GCs. The HB morphology of our sample of GCs has been investigated on the basis of the two new parameters L1 and L2 that measure the distance between the red giant branch and the coolest part of the HB and the color extension of the HB, respectively. We find that L1 correlates with both metallicity and age, whereas L2 most strongly correlates with the mass of the hosting GC. The range of helium abundance among the stars in a GC, characterized by ΔY and associated with the presence of multiple stellar populations, has been estimated in a few GCs to date. In these GCs we find a close relationship among ΔY, GC mass, and L2. We conclude that age and metallicity are the main global parameters, while the range of helium abundance within a GC is the main nonglobal parameter defining the HB morphology of Galactic GCs. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.