The "new" Gmail inbox has been around in some form for about a year. Teachers may have seen the inboxes show up in both the desktop and mobile versions of Gmail, but not given it too much thought.
Google well aware how we still struggle organizing our inbox decided to help with the organizing part. In examining our gmail for patterns, they came up with five core email message types; primary, social, promotions, updates, and forums.
In short, you now have five inboxes with the primary being the highest priority. It seperates your less important gmails into the other inboxes which you can get to as time permits. You can easily drag specific gmails to other inboxes if you think its a better categorization. The more you use this inbox feature the more accurate it will get.
Again a shoutout to +Sam Rosenfeld and his YouTube Playlist videos on using Gmail.