Full Idea
A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system.
Gist of Idea
A structure is a 'model' when the axioms are true. So which of the structures are models?
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Reference
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.280
A Reaction
This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key.