I present a computational framework for understanding the social aspects of emotions in Twitter conversations, where a conversation is a consecutive chain of reply tweets. Using that data, I explore in-depth questions of the emotional patterns in conversational interactions. I look for meaningful patterns of emotional exchanges in a conversation, and those patterns may depend on the topics and words of the conversation. I also hypothesize that conversational partners can influence each others` emotions and topics. Further, I discover interesting patterns in the overall emotions, affected by the lexical usages of the interlocutors. To find these patterns, I develop a novel computational framework, based on LDA, to discover the emotions from an unannotated corpus of Twitter conversations, and I evaluate the model by a human-annotated corpus. Specifically, I use LDA with Dirichlet Forest prior (DF-LDA) to discover emotion-centric topics from unannotated corpus. From the evaluation I verify our framework as a practical emotion classifier. I find that conversational partners usually express the same emotion, which I name Emotion accommodation, but when they do not, one of the conversational partners tends to respond with a positive emotion to a negative emotion rather than vice versa. I also show that tweets containing sympathy, apology, and complaint are significant emotion influencers. One interesting pattern is that Twitter users tend to feel happy at morning and feel angry at the night. Finally, I discover lexical patterns, such as the usage of swear words, that influence the overall emotion of a conversation.