Once you’ve beaten the final boss, you have the option to continue to play in the same 'world' as before or you can rest at the Majula bonfire and refresh the game again.
This time though the world changes slightly and there’s new enemies, new items and new surprises in store. Oh, and enemies hit much harder and have much more health this time too. But … you do get to keep all your weapons, stats and rings from before (you’ll lose all the keys and Fragrant Branch of Yores though).
Even at the very beginning there’s a - nasty - surprise in store and from them on it only gets more intense! There’s substantially more Dark Spirits here and you’re also much more likely to be invaded by a human-controlled opponent.
In many ways, New Game + is the real Dark Souls 2 and the first run was just a practice session. Good luck!