Each night in Terraria the moon moves across the sky from left to right (West to East). As each night passes, the moon moves into a different phase. Even if it is a Blood Moon, the cycle will not be disrupted, and continue as usual.
Including the Blood Moon, you could say there are actually 15 different phases of the moon, due to something such as a full moon becoming a full Blood Moon (a new moon cannot be a Blood Moon due to the moon not being in the sky). Without the Blood Moon there are only eight phases.
Plumbers Outfit (pants and shirt only) can only be bought from the Clothier on a full moon for 25 each.
The Doctor's Outfit can only be purchased from the Clothier on the night after a full moon for 20 each.
Werewolves only spawn on Full Moons, after the Wall of Flesh has been defeated in the underworld. Along with this, if the player was wearing a Moon Charm, they too transform into a werewolf during the full moon.
You can very faintly hear werewolves howl after a Full Moon.