The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions, given in a concrete way.
A multiplicity one theorem may also refer to a result about the restriction of a representation of a group G to a subgroup H. In that context, the pair (G, H) is called a strong Gelfand pair.
The group of adelic points of G, G(A), is said to satisfy the multiplicity-one property if any smooth irreducible admissible representation of G(A) occurs with multiplicity at most one in the space of cusp forms of central character ω, i.e. mπ is 0 or 1 for all such π.
The fact that the general linear group, GL(n), has the multiplicity-one property was proved by Jacquet & Langlands (1970) for n = 2 and independently by Piatetski-Shapiro (1979) and Shalika (1974) for n > 2 using the uniqueness of the Whittaker model.
The strong multiplicity one theorem of Piatetski-Shapiro (1979) and Jacquet and Shalika (1981a, 1981b) states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.