A structured set is, of course, a set equipped with extra structure. It is not the individual structured set that matters so much as the category of sets with a particular sort of structure.
Very abstractly, we may define a structured set as an object of any concrete category, that is an object of any category $C$ equipped with a faithful functor $U\colon C \to Set$ to the category of sets. (Some authors require that $U$ be representable for a concrete category, but we do not need that here.) Given two structured sets $X$ and $Y$ (in the same category $C$), a function $f\colon U(X) \to U(Y)$ between their underlying sets preserves the structure if it lies in the image of $U$, that is if there exists a (necessarily unique since $U$ is faithful) morphism $\tilde{f}\colon X \to Y$ in $C$ such that $U(\tilde{f}) = f$.
More concretely, we may define a type of structure on sets as an operation $T$ that, to any set $A$, assigns a set $T(A)$ of $T$-structures on $A$. At minimum, we should have $T(A) \cong T(B)$ whenever $A \cong B$ (where $\cong$ is isomorphism in $Set$), so that the concept is structural. But for good behaviour, we actually want something more coherent; we want an additional operation that, to any bijection $f\colon A \to B$, assigns a bijection $T(f)\colon T(A) \to T(B)$, such that:
In other words, $T\colon Set_\cong \to Set_\cong$ (or equivalently $T\colon Set_\cong \to Set$) is a functor from the underlying groupoid of $Set$ to itself (or equivalently to all of $Set$). In particular, any automorphism of a single set $A$ defines an automorphism of the $T$-structures on $A$, giving an action of the symmetric group $S_A$ on $T(A)$.
(Compare the notion of structure type from combinatorics, which is a set-valued functor on the groupoid of finite sets. Every combinatorial structure type can be interpreted as a type of structure, where only finite sets are capable of supporting the structure.)
Given a type $T$ of structure on sets, we define a $T$-structured set to be a set $A$ equipped with an element of $T(A)$. Given $T$-structured sets $X = (A,\sigma)$ and $Y = (B,\tau)$, a bijection $f\colon A \to B$ preserves the $T$-structure on $X$ and $Y$ if $T(f)(\sigma) = \tau$.
In general, there is no notion of whether an arbitrary function $f\colon A \to B$ preserves $T$-structure, although such a notion may be defined in many cases. So to get a concrete construction of a concrete category, we specify whatever morphisms we like, subject to the restriction that they form a category and have the correct core given above.
Bourbaki's theory of structure, while not described in category-theoretic terms, is essentially the above.
Morally, either of the abstract and concrete versions can be converted into the other. Technically, there are some restrictions.
Given a category $C$ and a faithful functor $U\colon C \to Set$, we may define a type $T$ of structure on sets as follows:
For each set $A$, consider the essential fibre of $U$ over $A$, the collection of pairs $(X,f)$ where $X$ is an object of $C$ and $f\colon U(X) \to A$ is a bijection. We consider two such pairs $(X,\sigma)$ and $(Y,\tau)$ to be equivalent if there is an isomorphism $h\colon X \to Y$ in $C$ such that $U(h) ; \tau = \sigma$. (Because $U$ is faithful, any such $h$ must be unique.) Define $T(A)$ to be the quotient set of the essential fibre modulo this equivalence relation. That is, a $T$-structure on $A$ is an equivalence class $[(X,\sigma)]$.
Given a bijection $f\colon A \to B$, we must define $T(f)\colon T(A) \to T(B)$. So given $(X,\sigma)$ as above, let $T(f)$ map $[(X,\sigma)]$ to $[(X,\sigma;f)]$ in $T(B)$. It's easy to check that this is well defined as a function from $T(A)$ to $T(B)$; we can also check that this makes $T$ into a functor and that the abstract and concrete definitions of whether a bijection preserves $T$-structure agree.
Technicality: If $C$ is a large category, then $T(A)$ might be a proper class instead of a set. In this case, we can pass to a larger universe; it is not essential for $T\colon Set_\cong \to Set$ that both copies of $Set$ be the same size. But for the above description to make sense as it is, we must require that $U$ have essentially small fibres.
Given a type $T$ of structure on sets, we cannot quite reconstruct the category $C$, but we can reconstruct its core $C_\cong$. That is, we can say what the objects and isomorphisms of $C$ are, if not the morphisms of $C$ in general.
An object of $C$ is simply a pair $(A,\sigma)$ consisting of a set $A$ and an element $\sigma$ of $T(A)$. Given two such objects, an isomorphism from $(A,\sigma)$ to $(B,\tau)$ is simply a structure-preserving map from $A$ to $B$, that is a bijection $f\colon A \to B$ such that $T(f)(\sigma) = \tau$. Then it is straightforward to check that this defines a groupoid $C_\cong$. This groupoid, the groupoid of $T$-structured sets, is naturally equipped with a faithful forgetful functor $U\colon C_\cong \to Set$, given by $U(A,\sigma) \coloneqq A$.
While defining isomorphisms of structured sets is an exact science, choosing more general morphisms of structured sets is something of an art. In principle, we may define a (not the!) category of $T$-structured sets by picking, for each $(A,\sigma)$ and $(B,\tau)$, a collection $Hom_{\sigma,\tau}(A,B)$ of functions from $A$ to $B$, such that:
whenever $f \in Hom_{\sigma,\tau}(A,B)$ and $g \in Hom_{\tau,\upsilon}(B,C)$, then $f ; g \in Hom_{\sigma,\upsilon}(A,C)$;
the identity map on $A$ belongs to $Hom_{\sigma,\sigma}(A,A)$; and
a bijection $f$ from $A$ to $B$ is an isomorphism (as defined above) if and only if both $f \in Hom_{\sigma,\tau}(A,B)$ and $f^{-1} \in Hom_{\tau,\sigma}(B,A)$.
The last condition states precisely that the underlying groupoid of any concrete category of $T$-structured sets is the groupoid of $T$-structured sets.
Any category of $T$-structured sets is still (like the groupoid of such sets) a concrete category.
If we start with a type $T$ of structures on sets, construct from this a groupoid $C_\cong$ and a faithful functor $U\colon C_\cong \to Set$ and then construct from this another type $T'$ of structures, then $T'$ will be equivalent to $T$ in the sense that there is a natural isomorphism between them as functors from $Set_\cong$ to $Set$.
If we start with a category $C$ and a faithful functor $U\colon C \to Set$, construct from this a type of structures $T\colon Set_\cong \to Set$, and then construct from this a groupoid $C_\cong$ with a faithful functor $U\colon C_\cong \to Set$, then $C_\cong$ will in fact be the core of $C$, with $U\colon C_\cong \to Set$ to restriction of $U\colon C \to Set$ to this core, up to equivalence of categories.
(Do we need proofs?)
Thus the abstract and concrete approaches to structured sets are equivalent, except that the concrete approach does not include a specification of what are the noninvertible morphisms between structured sets.
Almost everything in contemporary mathematics is an example of a structured set; here we list only a few representative ones (and perhaps also some exceptions).
Given any category $S$ whatsoever, we may define a type of structure on objects of $S$ as a functor $T\colon S_\cong \to Set$ to $Set$ from the underyling groupoid of $S$. Then any faithful functor $U\colon C \to S$ whatsoever defines a type of structure on objects of $S$ (at least if its fibres are essentially small), in the same way as a concrete category defines a type of structure on sets. Indeed, we say that $U$ presents the objects of $C$ as objects of $S$ with extra structure.