Inbox provides internal messaging within the system among users. Users receive system generated notifications as well as messages sent from other users in their inbox. They can compose and send messages to other users in the system as well. Messages can be sent to user or group. Inbox is an optional feature which can be turned ON from Site Details->Features tab. Note that these messages are not emails, they stay within the online system. Composed messages can contain rich text formatting, images and attachments.
Admins can send messages as email as well, compose message screen provides an option 'Send Email' if that's checked, message will be send to user as an email as well along with systems's inbox message.