Quantifier: "For all sets" I've seen the following statement a few times:

"Let $A$ be a set, then $\emptyset\subseteq A$".
Or, written 'more formally':
  $$
\forall A\,\, \emptyset\subseteq A
$$

My doubt is: I've always seen the quantifier $\forall x$ to mean "for all $x$ elements of some set $S$". However, when talking about all the sets, how do we define this quantification? 

$\forall A$ means "for all $A$" or more completely "for all objects $A$ that the theory we are working in is talking about". If the theory is set theory (say, we started off by writing down the axioms of ZFC), then our $A$ is a set; when working in other theories it may instead mean that $A$ is a natural number or that $A$ is a real number. However, this doesn't not happen by magic or even by convention, it happens by the very fact that we accept the appropriate axioms to hold for our thingies. 
So by the very fact that we work with  $\forall A\forall B\exists C\colon ((C\in A\leftrightarrow C\in B)\leftrightarrow A=B)$ and $\forall A\forall B\exists C\forall D\colon(D\in C\leftrightarrow (D=A\lor D=B))$ and so on as axioms, it happens that our thingies behave exactly as sets are supposed to behave.
