In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. In fact this is a normal way of proceeding; in the absence of recognisable structure (which might be hidden) problems tend to fall into the combinatorics classification of matters requiring special arguments.
In category theory structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structures. Starting with a given class of algebraic structure, such as groups, one can build the category in which the objects are groups and the morphisms are group homomorphisms: that is, of structures on one type, and mappings respecting that structure. Starting with a category C given abstractly, the challenge is to infer what structure it is on the objects that the morphisms 'preserve'.
The term structure was much used in connection with the Bourbaki group's approach. There is even a definition. Structure must definitely include topological space as well as the standard abstract algebra notions. Structure in this sense is probably commensurate with the idea of concrete category that can be presented in a definite way - the topological case means that infinitary operations will be needed. Presentation of a category (analogously to presentation of a group) can in fact be approached in a number of ways, the category structure not being (quite) an algebraic structure in its own right.
The term transport of structure is the 'French' way of expressing covariance or equivariance as a constraint: transfer structure by a surjection and then (if there is an existing structure) compare.
Since any group is a one-object category, a special case of the question about what the morphisms preserve is this: how to consider any group G as a symmetry group? That is answered, as best we can by Cayley's theorem. The analogue in category theory is the Yoneda lemma. One concludes that knowledge on the 'structure' is bound up with what we can say about the representable functors on C. Characterisations of them, in interesting cases, were sought in the 1960s, for application in particular in the moduli problems of algebraic geometry; showing in fact that these are very subtle matters.