Assignment refers to the function which assigns a reference to a variable. The truth-value of quantified formulas is determined with respect to a given assignment.
In order to determine whether All(v)[P] is true with respect to an assignment g we have to check whether P is true for every assignment g* differing at most from g in the value it assigns to v. Exists(v)[P] is true with respect to an assignment g when P is true for at least one assignment g* which may only differ from g in the value it assigns to v.
The use of assignments to variables makes it possible to interpret multiply quantified formulas in a compositional way.
Gamut, L.T.F. 1991. Logic, language, and meaning. Chicago: Univ. of Chicago Press.