There are four main factions shown in TFA:
The New Republic - formed from what was the Rebel Alliance, since they are no longer rebelling, they have won.
The planet that gets destroyed by Starkiller base is the New Republic's current senate headquarters (the senate rotates planets, like the Olympics)
The Empire - still controls some systems, but is at peace with the Republic now and has been supposedly de-militarized. (not directly shown in the movie)
The First Order - formed from remnants of the Empire who didn't want to give up fighting, they operate illegally from the shadows.
The Resistance - Princess Leia left the Republic to fight the First Order, but officially doesn't have Republic support (unofficially is perhaps a different matter). EDIT: To clarify further, this is therefore a new faction, with many former Rebel Alliance members in it perhaps, but with a different goal than the Rebel Alliance (defeat First Order, instead of restore the Republic), hence the new name.
(Source: Star Wars: The Force Awakens Visual Dictionary)