This article serves as an introduction to the study of networks of social systems. First, we introduce the reader to key mathematical tools to study social networks, including mathematical representations of networks and essential terminology. We describe several network properties of interest and techniques for measuring these properties. We also discuss some popular generative models of networks and see how the study of these models provides insight into the mechanisms for the emergence of structural patterns. Throughout, we will highlight the patterns that commonly emerge in social networks. The goal is to provide an accessible, broad, and solid foundation for a reader who is new to the field so that they may confidently engage more deeply with the mathematical study of social networks.