I apologize if my question is not well formed. The reason for it is that I don't understand the concepts enough to be able to ask a completely meaningful question.
In the classes we said that a theory $T$ consists of a set of logical symbols and statements which we call axioms.
Edit: I am only talking about propositional logic here.
We say that a logical statement $A$ can be proven by $T$ if there is a proof of $A$ using the standard deduction rules for logic and assuming the axioms of $T$.
Next we defined the model $M$ of a theory $T$.
Given the tuple $(B,v)$ where $B$ is a Boolean algebra and $v:X \mapsto B$ is the so called valuation function mapping symbols $X$ to elements of $B$.
We defined the meaning of a statement $A$ as $[A] \in B$ in the natural inductive way
$[\perp] = 0$, $[\top] = 1$
$[p] = v(p)$ if $p$ is a logical symbol
$[A \vee B] = [A] \vee_B [B]$ (here $\vee_B$ denotes the respective operation asociated with $B$)
and so on.....
We then said that a statement $A$ is valid if $[A]= 1$ so the meaning of $A$ wrt. a model $M$ is the element $1$ of the Boolean algebra in $M.$
A model of the theory $T$ is any $(B,v)$ in which the axioms of $T$ have meaning 1.
After that, many theorems followed relating the validity of a statement $A$ in the model $M$ of a theory $T$ and the provability of that statement in $T$.
Here is where I get lost. What exactly is going on? Why are we even introducing models? This new level of abstraction when dealing with theories confuses me completely. How do this models arise in dealing with "practical" mathematical proof working? Is there a dumb example?