Instead of rolling for initiative, at the start of each round everyone will declare their acting order, depending on the order picked, you're gonna get one benefit or the other.
Fast goes first, PCs going fast get to act immediately, before anything else.
Slow goes after, slow order seeks to represent those PCs that took some time to analyze the battleground, and because of this, they can impose Advantage or Disadvantage to one roll.
NPCs also follow a fast/slow system, however, its more straightforward. the weak or regular enemies will always go fast, they won't change their order ever. And contrary to this, big bad enemies will always go slow.
Enemies also benefit from Advantage and Disadvantage.
Death Saves go first, even before fast, they happen at the top of the round.
These work the same, your reaction is given back to you at the start of your next turn and abilities end when they are supposed to.