Recent advances in deep learning typically, with the introduction of transformer based models has shown massive improvement and success in many Natural Language Processing (NLP) tasks. One such area which has leveraged immensely is conversational agents or chatbots in open-ended (chit-chat conversations) and task-specific (such as medical or legal dialogue bots etc.) domains. However, in the era of automation, there is still a dearth of works focused on one of the most relevant use cases, i.e., tutoring dialog systems that can help students learn new subjects or topics of their interest. Most of the previous works in this domain are either rule based systems which require a lot of manual efforts or are based on multiple choice type factual questions. In this paper, we propose EDICA (Educational Domain Infused Conversational Agent), a language tutoring Virtual Agent (VA). EDICA employs two mechanisms in order to converse fluently with a student/user over a question and assist them to learn a language: (i) Student/Tutor Intent Classification (SIC-TIC) framework to identify the intent of the student and decide the action of the VA, respectively, in the on-going conversation and (ii) Tutor Response Generation (TRG) framework to generate domain infused and intent/action conditioned tutor responses at every step of the conversation. The VA is able to provide hints, ask questions and correct student's reply by generating an appropriate, informative and relevant tutor response. We establish the superiority of our proposed approach on various evaluation metrics over other baselines and state of the art models.