As more and more forms of AI become prevalent, it becomes increasingly important to understand how people develop mental models of these systems. In this work we study people's mental models of an AI agent in a cooperative word guessing game. We run a study in which people play the game with an AI agent while ``thinking out loud''; through thematic analysis we identify features of the mental models developed by participants. In a large-scale study we have participants play the game with the AI agent online and use a post-game survey to probe their mental model. We find that those who win more often have better estimates of the AI agent's abilities. We present three components---global knowledge, local knowledge, and knowledge distribution---for modeling AI systems and propose that understanding the underlying technology is insufficient for developing appropriate conceptual models---analysis of behavior is also necessary.