In section 4 of the article on Generalised Quantifiers in the Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/generalized-quantifiers/ the author writes:
"Modern predicate logic fixes the meaning of $\forall$ and $\exists$ with the respective clauses in the truth definition, which specifies inductively the conditions under which a formula $\phi (y1,…,yn)$ (with at most y1,…,yn free) is satisfied by corresponding elements b1,…,bn in a model $M = (M, I)$ (where $M$ is the universe and $I$ the interpretation function assigning suitable extensions to non-logical symbols): $M \vDash \phi(b1,…,bn)$. The clauses are
$M \vDash \forall x \psi(x, b1,…,bn)$ iff for each $a \in M$, $M \vDash \psi(a, b1,…, bn)$
$M \vDash \exists x \psi(x, b1,…, bn)$ iff there is some $a \in M$ s.t. $M \vDash \psi(a, b1,…, bn)$
To introduce other quantifiers, one needs to appreciate what kind of expressions $\forall$ and $\exists$ are. Syntactically, they are operators binding one variable in one formula. To see how they work semantically it is useful to rewrite (1) and (2) slightly. First, every formula $\psi(x)$ with one free variable denotes in a model $M$ a subset of $M$; the set of individuals in $M$ satisfying $\psi(x)$."
Why is it important that he write "(with at most y1,…,yn free)" and "every formula $\psi(x)$ with one free variable". Why should there be free variables?