I know there were a couple people above me who answered some of this, but when it comes to explanations I generally like more info than less so will add some in.

First thing to remember is that Delta only ever has 2 classes of service on its planes, whether international or domestic. This is different than other companies that may offer First/Business/Coach on a flight (or some variation on that name). The premium cabins on Delta are labeled First/Business Elite/Business depending on the situation.

First: This designation is used on the vast majority of flights domestically, in NA, and for some flights to SA. It generally comprises the services you are used to such as free booze, comfier and wider seats, occasionally in-flight power and food selections depending on the mileage.

Business Elite: The premium long-haul international cabin. Found with most (if not all) trips to Asia, Europe, Africa and some trips to SA. Provides better service than the domestic First class. Amenities vary but more and more cabins have lie-flat seats, you'll get a better selection of food items and many have AVOD.

Business: Has the exact same service as First. Business (sans Elite) is used for domestic flights that continue on to an international destination. You will have a domestic cabin and service (rare circumstances excepted). Perfect example is DL flight 36, goes LAS-CVG-LGW. The LAS-CVG section will be labeled Business, but it is on a domestically configured 737. The CVG-LGW section is on a internationally configured 757/767 depending on the time of year and will be labeled Business Elite.

Occasionally due to routing issues or plane movement you will end up on an internationally configured plane with a Business Elite cabin during a domestic run. My understanding is that these will be labeled as First/Business and will have standard domestic service (though will have comfier seats).