Discrete conformal structures on surfaces have been formulated primarily in two directions: (1) Thurston's formulation of circle packing and (2) discrete conformal structures coming from multiplicative weights on edges described by Luo and others. Both of these formulations have some key properties that provide tools for investigating the structures themselves and convergence of discrete conformal mappings to conformal mappings. Each determines a geometric structure on the Poincare dual of the geometric triangulation, leading to the formulation of a discrete Laplacian operator. In each case, the variation of curvatures with respect to vertex weights gives discrete Laplacians of the variation, in direct analogue to the Riemannian surface case. However, there are many other potential discrete conformal structures with this property. We will discuss the classification of all such discrete conformal structures. In particular, we can show that they are exactly those described by Zhang et. al. (2014). We will then discuss applications of this viewpoint to formulations of the discrete conformal mapping problem and recent results of Xu generalizing rigidity results for circle packing and multiplicative weight conformal structures.