Star Wars nerd time.
This is going to be hypothesis (obviously), but it'll be based on things that we do know definitively.
The death of Emperor Palpatine was pretty much the definitive fall of the Empire (at least as it was), but that doesn't mean that all of the Empire's remaining military forces suddenly blew up along with the second Death Star. They likely regrouped and took a new approach, since both of the Sith Lords that were leading them are/were dead. The First Order seems to be a new incarnation of those forces and the name also likely reflects the fact that it's being organized and lead by a new Order of collected individuals rather than a single Emperor. That's likely why/how they managed to maintain some degree of control over the Empire's territories after the death of Vader & Palpatine.
Being as it's unlikely that it's a New Jedi Order taking what the Old Republic had back (since our good guys still seem to be on the outskirts of the ruling powers of the galaxy), it's likely some of the remnants of the former Empire's power players pooling their resources into a new ruling body unlike any that's existed before -- hence them being The First Order. I'd expect this to mostly be made of individuals who were established military leaders like Admiral Thrawn & Grand Moff Tarken - though Kylo Ren makes it clear that there is still some Dark Side presence that appears to be working with them.
It's also likely that the way that The Order is implementing their changes doesn't sit well with everyone, hence the Rebellion's name change into the Resistance. This gives a sense that this group is made of the folks that are holding out against the reform that The First Order is establishing, rather than breaking away from an existing part of them. Those two names alone will tell you a lot about how the dynamics of the upcoming film and setting between The Resistance and The First Order will be both alike and different from the conflict between The Rebellion and The Empire.
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