Inbox is the place to look for activity that directly concerns you--notifications, mentions, direct messages, plus any activity on content you participated in. When there's a new reply to a discussion you participated in, you get a read alert here: when content you wrote gets a revision or a comment, you get a read alert for that too. The same thing happens when you get a mention, message, or system notification. You can choose to use alerts and read-tracking alone to keep up with your Inbox stream, or you can set your email notifications (under "[your name] > "Preferences") to send an email notification each time you get Inbox activity.
If you want activity you're not directly involved in to be tracked in your Inbox, just navigate to any item and click "Follow," selecting Inbox as the stream. You'll get alerted whenever there are changes or comments. If you decide to stop following it, just click "Stop following in Inbox" from the Inbox preview pane, or go to the item, click "Following," and uncheck Inbox. You won't be alerted about that item again.