In 1990, Kh. Al-Khnaifes, P. John and the author described some simple algorithms for calculating certain structural parameters (the number of Kekulé patterns, Pauling's bond orders, the number of spanning trees, the characteristic polynominal, the orbitals) of a benzenoid system. Such a system is a section of the hexagonal tessellation (or lattice) of the plane. In this paper, light is shed on the algebraic-combinatorial background. An infinite class of tessellations of the plane is defined, and it is pointed out that the same methods can be applied to sections of any tessellation of this class. Applications arise, e.g., in crystal physics in connection with the general dimer problem. Some remarks are made, and some first results are given, on (finite) hexagonal tessellations of the torus.