Many researchers studying online communities seek to make them better. However, beyond a small set of widely-held values, such as combating misinformation and abuse, determining what 'better' means can be challenging, as community members may disagree, values may be in conflict, and different communities may have differing preferences as a whole. In this work, we present the first study that elicits values directly from members across a diverse set of communities. We survey 212 members of 627 unique subreddits and ask them to describe their values for their communities in their own words. Through iterative categorization of 1,481 responses, we develop and validate a comprehensive taxonomy of community values, consisting of 29 subcategories within nine top-level categories, enabling principled, quantitative study of community values by researchers. Using our taxonomy, we reframe existing research problems, such as managing influxes of new members, as tensions between different values, and we identify understudied values, such as those regarding content quality and community size. We call for greater attention to vulnerable community members' values, and we make our codebook public for use in future research.