Your inbox shows all of your interactions in the community. If you, or another person create, respond, share etc in the community then it will be listed in your Boardville inbox. Additionally it will show when you've been tagged in content (called @mentioning), people liking your content (Acclaim) as well items like people following you. Simply select any item in your inbox and it will appear below where you can respond without going to the piece of content itself.
You can also opt out of content you are subscribed to by selecting the Hide new activity link to the bottom right of the above screen shot. This will stop any new notifications on the selected item being delivered to your inbox.