How can we model arguments and their dynamics in online forum discussions? The meteoric rise of online forums presents researchers across different disciplines with an unprecedented opportunity: we have access to texts containing discourse between groups of users generated in a voluntary and organic fashion. Most prior work so far has focused on classifying individual monological comments as either argumentative or not argumentative. However, few efforts quantify and describe the dialogical processes between users found in online forum discourse: the structure and content of interpersonal argumentation. Modeling dialogical discourse requires the ability to identify the presence of arguments, group them into clusters, and summarize the content and nature of clusters of arguments within a discussion thread in the forum. In this work, we develop ArguSense, a comprehensive and systematic framework for understanding arguments and debate in online forums. Our framework consists of methods for, among other things: (a) detecting argument topics in an unsupervised manner; (b) describing the structure of arguments within threads with powerful visualizations; and (c) quantifying the content and diversity of threads using argument similarity and clustering algorithms. We showcase our approach by analyzing the discussions of four communities on the Reddit platform over a span of 21 months. Specifically, we analyze the structure and content of threads related to GMOs in forums related to agriculture or farming to demonstrate the value of our framework.