A model is proposed for a class of two-person games based on the occupation of positions on a board. Well-known games that can be rephrased within or with the help of the model are, for example, tic-tac-toe, qubic, go-moku, hex, and connect-four. The model is formulated in terms of a composite algebraic structure called board. A notion of board isomorphism is defined, a few concepts fundamental for positional board game playing are identified, and necessary and sufficient conditions for establishing the isomorphism of two boards are proved. The formalism of the model provides criteria for the description and analysis of a board that are more abstract than its physical characteristics such as size and dimensionality. The application of the isomorphism results to the implementation of more general and efficient software modules for game playing is discussed. Related work is briefly outlined and compared.