Suppose we have a formula $\varphi(x,p_1,\dots,p_n)$. We want to say that all the elements which satisfy this formula are members of a certain set $S$. If we had only a formula with a single free variable $x$, we couldn't really say much about what this $x$ is like. But if we have additional parameters like $p_1,\dots,p_n$, we can say a lot about the "relations" between them and $x$, and we describe them using logical connectives and predicates proper to the language of First-Order set theory. So, the statement that an object $x$ is an element of $S$ given $x$'s relations to objects $p_1,\dots,p_n$ would be such ($\square$ here stands for either $\forall$ or $\exists$).
$$ \square S \square x\square p_1\dots\square p_n \, (x \in S \leftrightarrow \varphi(x,p_1,\dots,p_n)) $$
So, my understanding is that parameters are just free variables which help the formula express all the properties we want $x$ to have. The only thing that worries me is the order of the quantifiers $\forall$ and $\exists$, as I know that different arrangements of them produce a different meaning of our sentences. Maybe it should be $\exists S \forall x$? I don't know, that's my intuition. Please, do tell me in your answer.
So, have I correctly grasped what parameters are and what they are used for?
