After finishing the game for the first time, a new NPC will appear in the starting saloon area, with 3 Wanted Posters on the wall next to her. Interact with these posters to learn which areas the Outlaws are in for your current run.
When in an area with an Outlaw, the Marshal (your character) will make a comment when approaching the room they're in.
Outlaws are mini-bosses that function similarly to the Preacher fights: they use cover, have 2 different guns, throw explosives, and can't be stunned by lanterns.
The 2 mandatory Outlaw fights that take placed after the third area (Mines/Canyons) and fifth area (Town/Churchyards) do NOT count, only the 3 random "Wanted" Outlaws count.