An operator in predicate logic specifying for which values of a variable a formula is true. Universally quantified means "for all values" (written with an inverted A, LaTeX \forall) and existentially quantified means "there exists some value" (written with a reversed E, LaTeX \exists). To be unambiguous, the set to which the values of the variable belong should be specified, though this is often omitted when it is clear from the context (the "universe of discourse"). E.g.
Forall x . P(x) <=> not (Exists x . not P(x))
meaning that any x (in some unspecified set) has property P which is equivalent to saying that there does not exist any x which does not have the property.
If a variable is not quantified then it is a free variable. In logic programming this usually means that it is actually universally quantified.
See also first order logic.