Basic Economy - even shittier than Economy, you won't earn miles, can't pick your seat, and no changes are permitted
Main Cabin - Virgin America has used this phrasing for their Economy Class for a long time now, I wonder if that had any influence on Delta's choice here.
Delta Comfort+ - I think Premium Economy has a different name on every single airline. Again, they're copying Virgin in that they're including free drinks, better food, and free movies along with the extra 4" of legroom. Let's hope they don't copy Virgin's ridiculous pricing to buy these seats.
First Class - The standard USA domestic first you've seen for years. 38" recliner seats, usually with free drinks and a meal. This is what I commonly refer to as "Crappy Domestic First". Delta currently refers to this as Business Class. Unlike most other airlines, Delta prices their Crappy Domestic First reasonably, so hopefully this change won't make it hard to know what I'm getting when I price shop on Kayak.
Delta One - The new name for their Fancy First Class. This is usually on long-haul flights to Asia and Europe, as well as flights between JFK and SFO/LAX. If you're flying to Latin America, South America, or Hawaii, make sure you check Seat Guru... I'm guessing Delta One will only be on a small handful of flights to those markets.