As an essential characteristic of the smart grid, energy demand users are being transformed from passive roles to active decision-makers. To analyze their decision-making behaviors, game theory has been widely applied on the demand side. This paper focuses on the classification and in-depth analysis of recent studies that propose game-theoretic approaches for decision optimization of multiple demand users. This analysis classifies scenarios into various game participant categories, including distributed energy prosumers, small-and middle-sized users, and large energy consumers. The in-depth analysis of each scenario, covering non-cooperative game, cooperative game, Stackelberg game, Bayesian game, and evolutionary game, is conducted by analyzing market operation mechanisms, model assumptions/formulations, and solution methods. Based on a comprehensive review of such studies, it is concluded that game-theoretic applications on the demand side can benefit both the grid and the users, e.g., reductions in the peak-to-average ratios and energy costs of the users. The prospects for the applications of game theory on the demand side are discussed, including application scenarios and methodologies. The overview presented in this paper is expected to support researchers in comprehending typical game-theoretic concepts, keeping with the latest research developments, and identifying new and innovative applications for the energy demand side.