'Conversing' means talking. In daily life, it's better to say 'chatting'. It's more informal than 'conversing'. 'Communicating' means sharing information, not just by talking, but also by body language, etc. It also means sharing information to help someone else understand. For example, if two people are mad at each other, someone can ask them, "Why don't you communicate with each other?"

Converse is the act of starting a conversation, of starting a "talk" or speaking to others. Communicate is sharing information with a person in different ways that can be understood, including speech, sign languages, body languages, whistles. Animals can communicate, but they cannot converse.