The common distinction is between syntactical truth and semantic truth. Given a deduction system (i.e., some rules telling us what which strings are allowed and how to deduce new the syntactical truth of a sentence given the syntactical truth of others) you get a well-defined notion of syntactic truth as those statements that are derivable in the deduction system from a given theory.
In contact, semantic truth relates to a given model and is means those statements that when interpreted in $M$ are true.
Goedel's completeness theorem states that (for first order logic) in a deduction system, a statement is syntactically true given a theory $T$ if, and only if, it is semantically true in all models of $T$. So, this answers your question.
The fact that syntactic truth implies semantic truth is quite easy to prove. The other direction is involved and requires a slightly weaker axiom than the axiom of choice.