This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two distinct vertices with different orders are adjacent if the order of one vertex divides the order of the other. We prove some beautiful results in order-divisor graphs of finite groups. The primary focus is on examining the girth, degree of vertices, and size of the order-divisor graph. In particular, we provide a comprehensive description of these parameters for the order-divisor graphs of finite cyclic groups and dihedral groups.