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An abstract structure is a formal object that is defined by a set of laws, properties, and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fields that apply them, as computer science, and in the studies that reflect on them, as philosophy and especially the philosophy of mathematics. Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure).
An abstract structure may be represented (perhaps with some degree of approximation) by one or more physical objects - this is called an implementation or instantiation of the abstract structure. But the abstract structure itself is defined in a way that is not dependent on the properties of any particular implementation.
Example — the rules of chess
The rules of chess are an abstract structure, because their definition is independent of any particular chess set or board or chess notation. In this abstract structure, the king, for example, is defined as a piece that can move one square in any direction (except that it may not move onto a square that is under attack by an enemy piece). The king is not defined as a tall piece with a small cross on top, because it could be represented instead by the letter K, a particular sound frequency, a computer icon, or a small figurine of Cerebus the Aardvark. Because chess is an abstract structure, it is possible (and not unusual among chess masters) to play a game of chess that is entirely mental. (This might seem to require an exceptional memory, but in fact an exceptionally good understanding of the game obviates that need.)
Other board games such as checkers/draughts and go are also examples of abstract structures. Most sports, on the other hand, are not abstract structures because their rules depend on the physical properties of the pitch, ball or other playing equipment.
An abstract structure has a richer structure than a concept or an idea. An abstract structure must include precise rules of behaviour which can be used to determine whether a candidate implementation actually matches the abstract structure in question. Thus we may debate how well a particular government fits the concept of democracy, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess.
Other examples
A sorting algorithm is an abstract structure, but a recipe is not, because it depends on the properties and quantities of its ingredients.
A simple melody is an abstract structure, but an orchestration is not, because it depends on the properties of particular instruments.
Euclidean geometry is an abstract structure, but the theory of continental drift is not, because it depends on the geology of the Earth.
A formal language is an abstract structure, but a natural language is not, because its rules of grammar and syntax are open to debate and interpretation.
See also
"Abstract structure"