The main objective of this research is to examine the role of power relationship in a two-level green supply chain which is made up of one shared manufacturer and two competitive retailers. We develop six game theory-based models to explore the members’ operational decisions in a supply chain taking into account three vertical power structures (Manufacturer Stackelberg, Retailer Stackelberg, and Vertical Nash) as well as two retailers’ horizontal power structure (Bertrand or Stackelberg competition). Then, we design a two-part tariff contract which can encourage the supply chain members to promote cooperation and eventually coordinate the decentralized green supply chain under each power structure. Lastly, to further discuss the impact of the green awareness of consumers and the greening cost on supply chain players’ operational decisions and profits, we employ some numerical examples to conduct sensitivity analysis. The main conclusions are as follows. Firstly, the impact of power structure on the supply chain players’ operational decisions and profits mainly depends on the substitutability of the green products, the green awareness of consumers, and the greening cost for the manufacturer. Secondly, the more power the manufacturer has, the lower product greenness will be set. Thirdly, the consumer’s environmental awareness (the greening cost) positively (negatively) influences the manufacturer’s product greenness and wholesale price, the retailers’ sales prices, and the player’s profits under each power structure. Finally, the developed two-part tariff contract is practicable and beneficial for both the manufacturer and the two retailers.