Each NPC has their own requirements.
Merchant needs you to have collected 50 Silver Coins
Nurse needs you to have found a Crystal Heart
Demolitionist needs you to have found a Bomb or Dynamite
Arms Dealer needs you to have acquired a Musket (which is dropped from a Shadow Orb)
Dryad needs you to have killed a boss
Clothier needs you to have killed the Skeletron
Also, they all need their own house.
A proper house is:
At least 5 blocks high, 7 blocks wide building
No dirt
A backwall
A Table/Work Bench
A Chair
A light source
A door
