In mathematical logic a sharp distinction is made between the syntax and the semantics associated with the particular formal language that is being studied. The syntax is generally nothing but rules of how to combine the symbols in the logic. For example you can write "a+b" but not "+a b".
(It's important to note that you might want to allow "+a b", but that is what having different languages with different rules of combining the symbols is all about. In the current context this implies that a language that uses infix notation is different from a language that uses postfix or prefix notation.)
The semantics associated with this language or syntax (or logic) is generally some class (i.e. collection of) that gives "concrete" examples of the syntax. These concrete examples are generally some class of algebraic structures. For example the standard formal logic we use, also refered to as Boolean or propositional logic, has Boolean algebras as underlying semantics.
If we add the equality symbol, the universal and existential quantifiers and first order predicates to Boolean logic we get what is called first order logic (c.f. also predicate logic). Model theory is the study of the semantics, hence the structures, relating to first order logic. These structures are sometimes referred to as models of the language, hence the terminology. Historically the term came into common use midway through the 20th century. (Two seminal books on this subject are "Model Theory" by W. Hodges  and a book by the same title authored by C. Chang and H. Kiesler .)
Closely related fields include algebraic logic and universal algebra.