How does model theory connect language with possible interpretations?

In model theory, the connection between language and possible interpretations is established through the concept of a structure (or model), which provides a formal realization for a given language.12

A first-order language consists of symbols for relations, functions, and constants. Model theory connects this language to reality by defining a structure $𝔄$—a mathematical entity that assigns specific objects and functions to those symbols. A structure acts as a model for a theory when the "distinguished" or "true" sentences of that theory are satisfied by the structure.12

This relationship is defined by the relation of satisfaction: a sentence is true in a structure if the interpretation of its symbols within that structure aligns with the sentence's requirements. This approach allows logicians to study how different interpretations (models) can satisfy the same set of axioms, leading to insights such as the existence of nonstandard models. Furthermore, this semantic framework provides a rigorous way to define truth conditions for formal languages, as pioneered by Alfred Tarski, by specifying the conditions under which a sentence holds true within a given model.234