In this article, we study the activity patterns of modern social media users on platforms such as Twitter and Facebook. To characterize the complex patterns we observe in users interactions with social media, we describe a new class of point process models. The components in the model have straightforward interpretations and can thus provide meaningful insights into user activity patterns. A composite likelihood approach and a composite EM estimation procedure are developed to overcome the challenges that arise in parameter estimation. Using the proposed method, we analyze Donald Trumps Twitter data and study if and how his tweeting behavior evolved before, during and after the presidential campaign. Additionally, we analyze a large-scale social media data from Sina Weibo and identify interesting groups of users with distinct behaviors; in this analysis, we also discuss the effect of social ties on a users online content generating behavior.