Chemical reaction network theory is a field of applied mathematics concerned with modeling chemical systems, and can be used in other contexts such as systems biology to study cellular signaling pathways or epidemiology to study the effect of human interaction on the spread of disease. This thesis seeks to understand a chemical reaction network's equilibrium points through the lens of algebraic geometry; in particular, by computing the positive part of the steady-state variety defined by polynomial equations arising from the assumption of mass-action kinetics. While some (restricted) techniques currently exist to fully understand the ideal defining the positive steady-state variety, this computation presents a significant challenge in general. This thesis provides the necessary and sufficient conditions for 2-reaction networks to produce a nonempty positive steady-state variety, and a systematic classification of positive steady-state varieties produced by at-most-bimolecular 1-, 2-, and 3-species 2-reaction networks, grounded in combinatorial and algebraic properties. The classification theorems in this thesis provide simpler criteria than previous research, and aim to provide a foundation for future analysis of larger networks.