Models and Interpretability Is there any theorem that states that a theory $T'$ is interpretable in a theory $T$ if a model of $T'$ can also be made a model of $T$ or something like this? 
What I'm asking is that is there a way one can infer interpretability of $T'$ in $T$ knowing something about their models?
 A: You should compare Tarski's definition of interpretability with the standard definition in model theory (see, e.g. Section 4.3 of A Shorter Model Theory by Hodges). 
The definition is a bit complicated, so I'm not going to reproduce it here. But an interpretation of a theory $T'$ in a theory $T$ does correspond to a way of turning models of $T$ into models of $T'$ (note the reverse of direction) as described here: Interpretation (model theory). That is, $T'$ is interpretable in $T$ if and only if for every model $M\models T$, there is a model $M'\models T'$ such that $M'$ is interpretable in $M$, and these interpretations are uniform, in the sense that each of the definable sets specified in the definition are defined by the same $L$-formula in each model $M\models T$. 
Tarski's definition as you reproduce it in the comments is stricter than the general model-theoretic definition on two ways. First, while the general model-theoretic definition allows the domain of the interpreted structure to be a quotient of a definable set in the original structure by a definable equivalence relation, Tarski requires the domain of the interpretable structure to be the same as the domain of the original structure. Second, Tarski requires the definitions (the set $D$) to be recursive. So in modern language, we might say that $T'$ is interpretable in $T$ in the sense of Tarski if $T'$ is a reduct of a recursive expansion by definitions of $T$. This is stronger than saying that $T'$ is interpretable in $T$, so it still corresponds to a way of turning models of $T$ into models of $T'$. 
Namely, $T'$ is Tarski-interpretable in $T$ if and only if there is a recursive set of definitions for the symbols in $L'$ in terms of the symbols in $L$, such that if we take any model of $T$ and turn it into an $L'$-structure by using these definitions, we get a model of $T'$. 
