In Terraria, there are bound NPCs: the mechanic, the wizard and the goblin tinkerer. They spawn bound underground, you can unbind them by talking to them and they will move into an available house. I've managed to find the first 2 easily, but the goblin tinker keeps eluding me. What are the best places to look? If they spawn and I miss them, will they spawn elsewhere, or will I need to scour the entire map again to find the spot where they spawned first?
The bound NPCs spawn just like monsters: They appear in a radius around you, off-screen, and will despawn at some point.
If they are killed or despawn by themselves before you get to them, they will eventually spawn around you again.
All of them will only spawn while you are underground. Additionally:
The Goblin Tinkerer has pretty much no preferred hangouts. See this answer I gave here; basically, the most effective way to find him is to wander around aimlessly underground while chugging Hunter Potions (it reveals bound NPCs just like monsters).
Playing fullscreen at native resolution also helps.
I think that the goblin tinkerer spawns more often in the jungle and the dungeon. You have to beat at least one goblin army first. For more info look here