"Appear offline allows you to set your status to show you as “offline” to your Blizzard friends while remaining connected to Battle.net. This means you can launch the app and game away without distraction.
Appearing offline will show you as offline to everyone in your Blizzard friends list. Once you have joined a game, the experience of appearing offline might be slightly different depending on which game you are playing. In the case of World of Warcraft, your guildmates will see your character come online and enter WoW’s in-game chat channels, and anyone who has you as a character-level friend will see you online on their friends list. Everyone outside of the same game as you will not see you online or playing any games.
Do you see a friend who’s offline but you really want to send them a message while it’s on your mind? You can now send offline friends messages and they will receive them the next time they come online! While appearing offline you can still send and receive messages in the app, but it is up to you when you want to respond."
There are a few more features including profiles and more social features and if you want to read more about that head over to the Blizzard blog below.