We examine the way race and racial categories are adopted in algorithmic fairness frameworks. Current methodologies fail to adequately account for the socially constructed nature of race, instead adopting a conceptualization of race as a fixed attribute. Treating race as an attribute, rather than a structural, institutional, and relational phenomenon, can serve to minimize the structural aspects of algorithmic unfairness. In this work, we focus on the history of racial categories and turn to critical race theory and sociological work on race and ethnicity to ground conceptualizations of race for fairness research, drawing on lessons from public health, biomedical research, and social survey research. We argue that algorithmic fairness researchers need to take into account the multidimensionality of race, take seriously the processes of conceptualizing and operationalizing race, focus on social processes which produce racial inequality, and consider perspectives of those most affected by sociotechnical systems.