Terraria

SUITABLE HOUSE
You need a suitable house for each NPC.

Each NPC requires a suitable house to live in (10 blocks across and 6 high with a chair and table/workbench, chair, background walls and a suitable light source)

You can build them on top of each other to save space or spread them out. They can also be in separate locations, such as outposts, but it easiest to keep them all together in case you need to buy something from an NPC. You can assign any NPC to an open, suitable house. Several NPCs sell different items depending on what Biome they occupy.

Any NPC who dies (for whatever reason) will respawn after a while if the requirements are still met. You need all of them moved in to get the achievement. It will pop when the last NPC moves in, regardless of the final NPC.

QUICK SOLUTION

You can also just reassign an NPC to an open room if you have all of them present, such as in a friend/booster's world.

ALL NPCs as of 1.3 Fishing update (current 12/18/2015)

The Guide - appears with you at start of the game. Whenever killed, he comes back automatically
Merchant - appears when you alone or all players combined have wealth of more than 50 silver coins
Nurse - appears after you have used your first Heart Crystal to increase your max health
Demolitionist - appears when you find your first Explosive Item (i.e. Bombs, Dynamite, etc.) or have one in your inventory
Arms Dealer - appears when you have a Gun/Missile Weapon in your inventory.
Dryad - appears after you have killed ANY Boss
Old Man/Clothier - appears in front of the Dungeon. You must talk to him at night and offer to lift his curse. This makes Skeletron appear and once that boss is killed (presumably lifting the curse), the NPC will show up at your house. The Dungeon is made of bricks and colored blue, pink, or green near a Desert/Ocean Biome.
Mechanic - bound somewhere in the Dungeon after Skeletron is killed. You just need to talk to her to unbind her. She will sell her wares right there and move to an open room whenever it is ready.
Goblin Tinkerer - bound within the Dungeon, the Underground Jungle, or simply Underground after defeating the Goblin Invasion; same as Mechanic
Wizard - bound in Underground or Dungeon after you have beaten the Wall of Flesh and unlocked Hard Mode; same as Mechanic
Steampunker - appears after you kill a Mechanical boss (The Twins, Destroyer of Worlds, or Skeletron Prime
Dye Trader - appears after you obtain any type of dye material, such as Pink berries, red husk, etc.
Painter - appears after you have 3 NPCs assigned to rooms
Witch Doctor - appears after you defeat Queen Bee. Queen Bee is found in Jungle Hives after you destroy the baby inside or by using an Abeemination in the Jungle. He sells better stuff if his house is placed in a Jungle Biome.
Pirate - appears after you defeat the Pirate Invasion, which is an upgraded form of the Goblin Invasion. The Invasion can be started with a Pirate Map item. He sells better stuff if his house is placed in an Ocean Biome.
Party Girl - has a 9% chance to show up every day once the player has at least 8 NPCs assigned to rooms and there is a open room for her
Cyborg - appears after you defeat Plantera, which requires you to kill one of the mechanical bosses (The Destroyer, The Twins, and Skeletron Prime). This gives the message "The Jungle grows restless," which means a Plantera bulb has appeared in the Underground Jungle. The more Mechanical Bosses you kill, the more bulbs spawn. You must destroy a bulb to summon Plantera.
Truffle - He's a tough one to get:
Basically, you need a mushroom biome above the underground that is not corrupt/crimson. It should look like the ones in the Underground Jungle and has its own music. You can make an "artificial" one by making a row of mud blocks 2 high x 25 long. Repeat these rows three times with space for you to walk through (3 block spaces). You then plant mushroom seeds or you can use jungle grass seeds in the mud and then hit them with the clentaminator and dark blue solution. Add some background (you can use glowing mushroom) walls. Then seal it off on all sides, add a door, a table, chair, and light source to finish your new room/biome. Check to make sure the house is suitable and the Truffle will move in.

NPCs NOT REQUIRED for achievement
Travelling Merchant - He only shows up after you have 2 NPCs moved in and has a 20% chance to spawn for 12 hours game time. He sells rare and unique items and WILL NOT MOVE INTO A ROOM. If you exit the game and come back, he will not be there.

Angler - He spawns by an Ocean biome, where he is sleeping until you wake him up. Then he will move in if you have an empty suitable room. He will give you quests which involve catching him rare fish from different biomes, and he rewards you with various items.

Santa Claus - He only shows up after you kill the Frost Legion and 'Tis the Season (around Christmas time IRL)

Stylist - Spawns in Spider areas underground, and you must free her (just like the Mechanic). Once freed, she sells Hair Dyes and will move in.

If you have any notes, let me know and I will add them with credit before you vote negatively.

***Same as 360 version***

UPDATE - It is currently possible to unlock this achievement by joining the world of another player who has all the NPCs already.
Feel free to message me on Xbox if you would like to join my world! A Journey character and mic are required.

Both of the existing guides for this achievement are completely out of date. As such I will provide an up-to-date list of all the required NPCs below.

The first thing you will need is suitable housing for each of the 25 NPCs required.

Suitable housing must be 10 blocks in width and 6 blocks high and have a back wall. It must also contain:
1x Table / Workbench
1x Chair
1x Light source

Once these requirements are met it's time to recruit your NPCs. Below I will outline each NPC required and the requirements for recruiting them:

Pre-Hardmode

1. Guide - No requirements to spawn him, he will always be present at the start of each new world.
2. Merchant - The combined wealth of all present players must exceed 50 Silver Coins.
3. Nurse - One player must have more than 100 health, and the Merchant must also be present.
4. Demolitionist - A player has explosives in their inventory, the Merchant must also be present.
5. Dye Trader - A player has a dye or any item used to craft dye in their inventory, any four other NPC must also be present.
6. Angler - Found and spoken to in an Ocean biome.
7. Zoologist - At least 10% of the Bestiary (53 entries) has been filled.
8. Dryad - The Eye of Cthulhu, Eater of Worlds / Brain of Cthulhu, or Skeletron has been defeated.
9. Painter - Four other town NPCs are in the world.
10. Golfer - Found and spoken to in the Underground Desert.
11. Arms Dealer - A player has bullets or a gun that shoots bullets in their inventory.
12. Tavernkeep - Found and spoken to after the Eater of Worlds or Brain of Cthulhu has been defeated. He can be found unconscious in any Biome.
13. Stylist - Found and spoken to in a Spider Cave.
14. Goblin Tinkerer - Found and spoken to in the Cavern layer, after a Goblin Invasion has been defeated.
15. Witch Doctor - The Queen Bee has been defeated (Summoned by breaking the Larva in a Beehive found in the Underground Jungle).
16. Clothier - Skeletron has been defeated.
17. Mechanic - Found and spoken to in the Dungeon after Skeletron has been defeated.
18. Party Girl - 1/40 chance of spawning each day after any eight other NPCs are in the world.

Hardmode

1. Wizard - Found and spoken to in the Cavern layer.
2. Tax Collector - A Tortured Soul is transformed with Purification Powder in the Underworld.
3. Truffle - A house is available in an above-ground Glowing Mushroom biome during Hardmode.
4. Pirate - A Pirate Invasion has been defeated. A Pirate Invasion can be started by using a Pirate Map which can drop from any enemy in an ocean biome.
5. Steampunker - A mechanical boss (The Destroyer, The Twins, or Skeletron Prime) has been defeated.
6. Cyborg - Plantera has been defeated.
7. Princess - All other town NPCs are in the world (except for town pets, slimes and Santa Claus).

The following NPCs are not needed for the achievement:

• Santa Claus
• Nerdy Slime
• Elder Slime
• Squire Slime
• Clumsy Slime
• Surly Slime
• Cool Slime
• Mystic Slime
• Diva Slime
• Cat
• Dog
• Rabbit

As soon as the Princess moves in the achievement will unlock.

Below I will outline a few tips and tricks for easily obtaining some of the more difficult NPCs:

• When looking for NPCs that spawn in underground biomes, you should use Hunter Potions if possible. These potions will make all enemies glow red, and all non-hostile mobs glow green (This includes NPCs). This will make spotting certain NPCs underground in the low light far easier.
• When searching for NPCs in specific biomes such as the Stylist in Spider Caves, it helps to create a tunnel going through the longest / widest section of that biome and then simply run or fly through it. The NPC you are looking for will eventually appear in or around this tunnel.
• Playing on Journey will make finding certain NPCs considerably easier as you can increase the spawn rate of all mobs. This will also increase the frequency of NPCs appearing.
• Setting the Time Speed to x24 will increase the frequency of NPCs moving into your available housing.
• In order to check which NPCs are currently in your world, you can go to the housing menu found in the Equipment section of the inventory. This will show you all the NPCs currently alive in your world.
• As stated above, none of the newly added slimes are required for the achievement. The same is true for the pets you can adopt.
• Once an NPC's initial spawn requirements have been met they will be able to respawn without needing to meet them again in the event that they are killed, as long as a vacant house is still available. For instance, once the Arms Dealer has moved in after having had bullets in your inventory, you can safely discard the bullets and he will reappear again if he dies without needing to be holding bullets.