I don't really understand what a model is. I've studied a bit of first order logic, and I still don't understand what a model really is.

A model of a theory $T$ is an interpretation which assigns the value True to its sentences.

Ok, that's pretty clear, I guess.
But then I run into things like this:
Let $T$ be the axioms of the ordered fields, then $(\Bbb R,+,\cdot)$ is a model of $T$.
At a glance, this looks OK, I mean, $\Bbb R$ satisfies all the axioms. But why?
Well, if we take the standard construction of $\Bbb R$ as dedekind cuts, we can see that $\Bbb R$ does satisfy them, IF $\Bbb Q$ does.
So we have to go back to the construction of, $\Bbb Q$, and realize we've defined this as $\Bbb Q = (\Bbb Z \times \Bbb Z)/\equiv_1$ for a suitable equivalence relation $\equiv_1$, then again, $\Bbb Q$ satisfies the axioms if $\Bbb Z$ some others...
Finally we're back to how we define $\Bbb N$, we don't really have a step below this (like we had with $\Bbb R,\Bbb Q, \Bbb Z$), so we assume some stuff (like the Peano axioms)? 

I believe this is a problem: by the completeness/soundness theorem of FOL, we have that $T$ defined as before must be consistent, as it has a model.  However, isn't the consistence of $T$ dependend on the consistence of (in this case) Peano axioms? 
Wouldn't we require a model of P.A? 
I'm not sure I've made clear my confusions, as all of this is still pretty fuzzy in my head, if anything needs clarification, please leave a comment.
 A: Tarski analysed what it means to give a semantics to a formal language. One of his conclusions is that we need to separate the concept of the "object language" (the language whose semantics we want to define) from the concept of the "metalanguage" (the language we are going to use to formulate the definition of the semantics).
For the vast majority of mathematical logicians, the metalanguage is just the usual semiformal mixture of words and symbols that most mathematical discourse involves. For them a model of PA would be given by the natural numbers of everyday mathematical practice.
You could take a much more formal view and take something like ZF set theory as your metalanguage, but that would involve a great deal of unnecessary work for someone who is just interested in using formal logic to investigate mathematical structures or studying it for its own intrinsic intellectual interest. If you took this approach, a model of PA would be given by something like von Neumann's construction of the natural numbers in ZF.
Whichever approach you take, you get no guarantees of consistency, but that isn't really the object of the exercise. If you actually want to construct models for an object language, then you need to make some ontological assumptions in your metalanguage. However, to work with object language syntax, you need to reason about strings of symbols and that is not really different from reasoning about natural numbers. So if you reject PA (or some constructive approximation to it), you have already rejected mathematical logic.
See this SEP article about Tarski's ideas for more discussion and references.
