Inbox is used to receive the notifications about the activities of a project. When the tasks, posts, events, files, etc. you follow have new activities, there will be a number to the right of the inbox.
Click to open a message in the inbox to view its detail panel. Click the navigation buttons on top of the detail panel to track it in its original project.
You can mark all the unread messages as read, and can clear all the read messages. Hover over a message, and the "Delete" button will show up automatically to the right of the message for you to delete it.