What is the idea behind "representability" in a first order theory? I've been reading through Enderton's logic, this notion is introduced and is given special attention as it's said that they are crucial in the proof of incompletness theorems. 
I grasp the formal definition, but what idea does this notion try to capture and formalize?
I think that a relation is representable in a given theory means that we can express the condition of membership of this relation using a sentence in that theory. Is that intuition valid? 
 A: Yes, that's the intuition you want -- except for minor terminology quibbles. You represent a relation with a formula with free variables for each position in the relation, which is specifically not a "sentence".
Note that there's a difference between representing a relation in a language with a particular interpretation (for example the language of arithmetic interpreted in the integers), and representing a relation in a particular theory.
In the former case, it is enough that you have a formula that happens to have the right truth value when you evaluate it in the integers.
The latter case is more subtle. Here the relation we want to represent is a given relation between the actual integers, but representability requires the the theory can prove that the relation is true or (as appropriate) false whenever we insert numerals into our formula -- independently of any particular model or interpretation of it.
So, for example, a relation that is representable in PA will always be representable in the structure $(\mathbb N,0,1,{+},{\times})$, but the converse is not always the case.
