An abstract structure is a set of rules, properties and relationships that is defined independently of any physical objects. Abstract structures are studied in philosophy, computer science and 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 computer icon, or a small figurine of Captain Kirk. Because chess is an abstract structure, it is possible in principle to play a game of chess that is entirely mental (provided that you and your opponent have very good memories !).
Other board games such as 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.