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? 

There is nothing wrong with your statement, though I would insert some punctuation:
$\forall A: \emptyset \subset A$  
or 
$\forall A (\emptyset \subset A)$ 
In algebra and analysis, the quantified variable may be an arbitrary placeholder, an element of a set, a set or a function. 
If the universally quantified variable is an element $x$ of a set $S$, then the quantifier can be written as:
$\forall x \in S:\space \cdots$ 
or as 
$\forall x:[x\in S \implies\cdots$ 
Similarly if the universally quantified variable is a subset $x$ of $S$, then the quantifier can be written as:
$\forall x \subset S:\space\cdots$ 
or as 
$\forall x:[x\subset S \implies\cdots$ 
If the existentially quantified variable is an element $x$ of a set $S$, then the  quantifier can be written as 
$\exists x\in S:\space\cdots$ 
or as 
$\exists x: [x\in S \land \cdots$
