Both discussions and conversations are ways to communicate with people in the community:
Discussions are specific to a group and can be categorized by topic. Discussions within FCP Family Groups can be seen only by team members, while discussion in the Community Center can be seen by all.
Conversations are a way to have a private chat with anyone across the community regardless of whether they are in your group or not. This functionality works similarly to an Instant Messager. This is private to only those involved in the conversation. Conversations cannot be archived and should only be used for short conversation.