Permission to go into a place — the door, the building, the grounds. Signs still say “No admittance,” meaning you may not enter. It is a slightly old-fashioned or official noun; in everyday talk people more often say you can’t come in, or they use admission. Though actually, admission and admittance are cousins that even native speakers mix up: admittance leans toward physically entering a place; admission is the wider word — tickets, being accepted into a school, or admitting a mistake. Engineers later used admittance for a technical idea about electric current (a sense coined in 1887), but that is specialist talk. For most learners, the door-sign meaning is the one that matters.