The language is not necessarily the one described in the article. More common is to have a single binary function symbol.
That said, the symbols of the language are precisely that, symbols. In particular, $\forall$ and $\exists$ are symbols, they do not range over anything.
After we have described the language, and the axioms, which are certain strings of symbols, we obtain a theory $T$. We are interested in models of that theory. Suppose for simplicity that our language has a single binary function symbol $p$. A model $\mathcal{A}$ of our theory is a non-empty set $A$, together with an honest to goodness binary function $p_\mathcal{A}\colon A\times A \to A$ such that under the usual definition of truth, all the axioms of $T$ are true in the structure with underlying set $A$ and binary function $p_\mathcal{A}$.
The details of the definition of truth are a bit too lengthy to give here. Roughly speaking, we define truth in $\mathcal{A}$ of sentences $\varphi$ of our language by an induction on the complexity of $\varphi$. As a simple example, we say that $\varphi\land \psi$ is true in $\mathcal{A}$ if both $\varphi$ and $\psi$ are true in $\mathcal{A}$. Note that $\land$ is a formal symbol. This part of the definition of truth in a sense assigns meaning to the formal symbol $\land$. But in principle the syntax, which deals with uninterpreted formal symbols, is kept strictly separate from the semantics (models). That separation is not always strictly maintained, because the most interesting questions deal with the interaction between syntax and semantics. The link between the two is the definition of truth in a structure.
To partly answer your question, we look at the sentence
$\exists x\forall y (p(x,y)=y).$ The above axiom is intended to assert the existence of a left-identity, but it is just a string of symbols. For this axiom to be true in $\mathcal{A}$ means that there is an element $e \in A$ such that for all $a\in A$, $f_A(e,a)=a$.
Thus it is at the model stage that the formal symbols $\forall$ and $\exists$ are interpreted as working like quantifiers in the usual informal sense. The technical details of the definition of truth in a structure $\mathcal{A}$ take care of that. Since we are working in a specific model, the quantification is always over a completely specific set $A$, so the universe is always a specific set.
To sum up, the models of the first-order Theory of Groups are precisely the set-theoretic groups.