A structure in mathematics (also “mathematical structure”) is often taken to be a set equipped with some choice of elements, with some operations and some relations. Such as for instance the “structure of a group”. In model theory this concept of mathematical structure is formalized by way of formal logic.
Notice however that by far not every concept studied in mathematics fits as an example of a mathematical structure in the sense of classical first order model theory, described below. For instance a concept as basic as that of topological spaces fails to be a structure in the sense of classical model theory (see here).
In category theory there is a more flexible concept of structure, see there.
Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a structure for $L$, or “$L$-structure”, is a set $M$ with an interpretation for symbols:
if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$
if $f\in L$ is an $n$-ary function symbol, then $f^M:M^n\to M$ is a function
if $c\in L$ is a constant symbol, then $c^M\in M$
The underlying set $M$ of the structure is referred to as (universal) domain of the structure (or the universe of the structure).
Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is true in $M$ if $\phi^M$ is true. Given a theory $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$ (axioms), the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a model of $(L,T)$.
In model theory, given a language $L$, a structure for $L$ is the same as a model of $L$ as a theory with an empty set of axioms. Conversely, a model of a theory is a structure of its underlying language that satisfies the axioms demanded by that theory.
There is a generalization of structure for languages/theories with multiple domains or sorts, called multi-sorted languages/theories.
A class $K$ of structures of a given signature is an elementary class if there is a first-order theory $T$ such that $K$ consists precisely of all models of $T$.
There is a vast generalizations for higher-order theories (and more), see at abstract elementary class and metric abstract elementary class.
Every algebraic category whose forgetful functor preserves filtered colimits is the category of models for some first-order theory. The converse is false.
A detailed discussion of characterizations of categories of structures in the sense of model theory is in (Beke-Rosciky 11).
Every first-order language $L$ gives rise to a first-order hyperdoctrine with equality freely generated from $L$. Denoting this by $T(L)$, the base category $C_{T(L)}$ consists of sorts (which are products of basic sorts) and functional terms between sorts; the predicates are equivalence classes of relations definable in the language. The construction of $T(L)$ depends to some extent on the logic we wish to impose; for example, we could take the free Boolean hyperdoctrine generated from $L$ if we work in classical logic.
There is also a “tautological” first order hyperdoctrine whose base category is $Set$, and whose predicates are given by the power set functor
and then an interpretation of $L$, as described above, amounts to a morphism of hyperdoctrines $T(L) \to Taut(Set)$.
This observation opens the door to a widened interpretation of “interpretation” in categorical logic, where we might for instance generalize Set to any other topos $E$, and use instead $Sub \colon E^{op} \to Heyt$ (taking an object of $E$ to its Heyting algebra of subobjects) as the receiver of interpretations. This of course is just one of many possibilities.
Standard textbook accounts include
Wilfrid Hodges, section 1 of A shorted model theory, Cambridge University Press (1997)
Chen Chung Chang, H. Jerome Keisler, Model Theory. Studies in Logic and the Foundations of Mathematics. 1973, 1990, Elsevier.
Characterizations of categories of model-theoretic structures and homomorphisms between them (certain accessible categories) is discussed in
Online discussion includes