This paper is a comprehensive study on the classification of torsion-free nil rings of rank two. It covers the full classification of nil rings with decomposable torsion-free additive groups of rank two, and gives a detailed description of nil rings with indecomposable torsion-free additive groups of rank two. The criteria for the decomposition of torsion-free abelian groups are also discussed. The study builds upon the foundational work initiated by \textit{R. A. Beaumont} and \textit{R. J. Wisner} [Acta Sci. Math. 20, 105--116 (1959; Zbl 0092.03703)], which introduced key concepts and methods in the study of rank two torsion-free rings. Following research, particularly by Arnold, resulted in the publication of a monograph that further advanced the understanding of torsion-free rings and their additive groups. The results of this paper is separated into two sections (Section 2 and Section 3). A significant contribution to this paper is the proof that, if and only if it is the additive group of a nil ring, an indecomposable non-nil torsion-free abelian group of rank two supports only nilpotent rings. The authors also discuss the difficulty of classifying torsion-free rings and abelian groups using quasi-isomorphisms, a method that has been found to be theoretically elegant but of limited practical application. They criticize the insufficiency of classical construction methods for torsion-free abelian groups of finite rank, and emphasize the need for more precise approaches to study torsion-free rings. A comprehensive categorization of nil rings with decomposable torsion-free additive groups of rank two and an in-depth description of nil rings with indecomposable torsion-free additive groups of rank two are among the primary results presented in this work. The authors also point out directions for further research in this field and talk about how their discoveries affect the structure of rank two torsion-free nil rings. To sum up, the paper is an important addition to the area of classification of nil rings since it presents a comprehensive study of rank two torsion-free nil rings and lays the groundwork for future research into the structure and categorization of these rings as well as the abelian groups that they are related to.