We analyze the structural and thermodynamic properties of D-dimensional (D≥4), asymptotically flat or anti–de Sitter, electrically charged black hole solutions, resulting from the minimal coupling of general nonlinear electrodynamics to general relativity. This analysis deals with static spherically symmetric (elementary) configurations with spherical horizons. Our methods are based on the study of the behavior (in vacuum and on the boundary of their domain of definition) of the Lagrangian density functions characterizing the nonlinear electrodynamic models in flat spacetime. These functions are constrained by some admissibility conditions endorsing the physical consistency of the corresponding theories, which are classified in several families, some of them supporting elementary solutions in flat space that are nontopological solitons. This classification induces a similar one for the elementary black hole solutions of the associated gravitating nonlinear electrodynamics, whose geometrical structures are thoroughly explored. A consistent thermodynamic analysis can be developed for the subclass of families whose associated black hole solutions behave asymptotically as the Schwarzschild metric (in the absence of a cosmological term). In these cases we obtain the behavior of the main thermodynamic functions, as well as important finite relations among them. In particular, we find the general equation determining the set of extreme black holes for every model, and a general Smarr formula, valid for the set of elementary black hole solutions of such models. We also consider the one-parameter group of scale transformations, which are symmetries of the field equations of any nonlinear electrodynamics in flat spacetime. These symmetries are respected by the minimal coupling to gravitation and induce representations of the group in the spaces of solutions of the different models, characterized by their thermodynamic functions. Exploiting this fact we find the expression of the equation of state of the set of black hole solutions associated with any model. These results are generalized to asymptotically anti–de Sitter solutions.