Well we have to remember that TFA was the first film this series, so I’m sure we’ll learn more about the First Order and why it seems to be the main counterpoint to the New Republic. I assume that since they seem to be a very legitimate force that they have a great deal of resources and have established themselves as superior to the other remnants, and thus likely have planets in their command. They wouldn’t call themselves the Galactic Empire again, because they aren’t. The Empire was Palpatine’s thing, and he failed. Plus they don’t really seem to have an Empire yet. The Republic is definitely in charge right now, but I seem to remember Hux saying something about the illegitimacy of the Republic’s rule. Also, their logo is significantly different from the Empire’s.

To conjecture a little bit more, I assume the name “First Order” was likely a moniker chosen during what was likely a civil war between different factions of the Imperial Remnant. It’s possible that the First Order came out on top due to having the Force-sensitive Kylo Ren and (probably) Snoke, as well as Sith relics (like Vader’s mask) that connected them to the Galactic Empire, which would legitimize them. Again, it’s conjecture, but I assumed that by kidnapping it was meant that it was a form of conscription, wherein potential troopers were taken from their parents at birth (like Jedi) from member worlds, less that they snuck into Republic homes and snatched kids in the middle of the night.

As far as the Rebellion/Resistance goes, I assume that the Rebellion split after Palpatine was assassinated, with part of it becoming the New Republic in order to consolidate the gains made after Endor, while Leia’s faction just rolled over into the Resistance to keep up the fight against what was left of the Empire. They’re pretty much the same people doing the same thing.