Moderated Chatroom is a room where a message can only be published after it is approved by a chat moderator/administrator.
If a user sends message, it is first sent to admins for approval. As soon as the message is approved by any of the admins, it will be sent to rest of the users present in that room. If it is not approved, then it is deleted.
1) A user David, who is not admin, sends a message in a moderated chatroom. This is how David's screen will look:
Moderated Chatroom in iFlyChat
2) Now this message will be received by all chat admins in their chat window. They may either approve it or delete it. If it is approved, then it will be sent to all users. This is how chat admin's screen will look (the option to accept or decline message will come only when you hover over the message):