The richness that characterizes relationships is often absent when they are modeled using computational methods in network science. Typically, relationships are represented simply as links, perhaps with weights. The lack of finer granularity is due in part to the fact that, aside from linkage and strength, no fundamental or immediately obvious dimensions exist along which to categorize relationships. Here we propose a set of dimensions that capture major components of many relationships -- derived both from relevant academic literature and people's everyday descriptions of their relationships. We first review prominent findings in sociology and social psychology, highlighting dimensions that have been widely used to categorize social relationships. Next, we examine the validity of these dimensions empirically in two crowd-sourced experiments. Ultimately, we arrive at a set of ten major dimensions that can be used to categorize relationships: similarity, trust, romance, social support, identity, respect, knowledge exchange, power, fun, and conflict. These ten dimensions, while not dispositive, offer higher resolution than existing models. Indeed, we show that one can more accurately predict missing links in a social graph by using these dimensions than by using a state-of-the-art link embeddedness method. We also describe tinghy.org, an online platform we built to collect data about how social media users perceive their online relationships, allowing us to examine these dimensions at scale. Overall, by proposing a new way of modeling social graphs, our work aims to contribute both to theory in network science and practice in the design of social-networking applications.