Terraria (UP)

NOTE: You must kill each slime with a weapon from your hands. Using Bunny Cannons, dynamite, etc. or having a friend kill them in MP does not count! However, it appears that minions, or star veils causing the death blow does count.

Each slime increases your % in the app by bout 5% if you are tracking it, but there is no way of knowing which ones you have killed without keeping track yourself.

NORMAL or HARD MODE Slimes
1. Green Slime - Common near surface (ANY Biome)
2. Blue Slime - Common near surface (ANY Biome)
3. Red Slime - Common near surface (ANY Biome)
4. Purple Slime - Common near surface (ANY Biome)
5. Yellow Slime - Common near surface (ANY Biome)
6. Black Slime - Common ONLY found underground
7. Mother Slime - Common ONLY found underground
8. Baby Slime - Common ONLY found underground; Spawns from Mother Slime when killed
9. Pinky - VERY RARE near surface or Underground (ANY Biome)
Suggested: Create New Worlds, search surface, and/or dig down to underground stone layer just before the underworld to increase chance of one spawning (credit to Crimson Drifter & sonnyforple)
Also use a Battle Potion to increase spawn rates and hold or place a water candle around you. You can also Exit without Saving and keep entering the New World to search the surface and use a Battle Potion until you find one. Best guess is that Pinky spawns in place of another slime, so wherever slimes can be found, there is a chance Pinky can show up. More slimes spawn on the surface of a newly created world, so that is the best place to get one to spawn.
10. Jungle Slime - Common ONLY found in surface of Jungle Biome
11. Spiked Jungle Slime - Uncommon ONLY found in Jungle Biome; more common in Underground Jungle
12. Lava Slime - Common ONLY found in Underworld
13. Dungeon Slime - Uncommon/Rare ONLY found deep in Dungeon
14. Ice Slime - Common ONLY found in Snow Biome
15. Frozen Spiked Slime - Uncommon ONLY found in Snow Biome; more common in Underground Snow areas
16. Umbrella Slime - Common near surface (ANY Biome) but ONLY during Rain Event
17. King Slime - Rare spawn rate boss; also summoned with Slime Crown (much easier method and Required for Exterminator achievement)
SLIME CROWN RECIPE
99 Gel
1 Gold/Platinum Crown - made from 30 Gold/Platinum bars and 1 Ruby

HARD MODE ONLY
18. Crimslime - Common near surface or Underground of Crimson Biome
19. Corrupt Slime - Common near surface or Underground ONLY in Corrupt Biome
20. Slimeling - Common near surface or Underground ONLY in Corrupt Biome; spawns from Corrupt or Shadow Slime when killed
21. Slimer - Common near surface or Underground ONLY in Corrupt Biome; has demon-like wings knocked off when hit
22. Shadow Slime - Uncommon near surface; more common in Underground ONLY in Corrupt Biome; looks similar to Corrupt Slime
23. Toxic Sludge - Common near surface or Underground ONLY in Hallowed Biome
24. Illuminant Slime - Common ONLY in Underground Hallow and glows neon pink
25. Rainbow Slime - Rare spawn in surface of Hallow ONLY during Rain Event; slightly larger than other slimes and glows rainbow colors

Pinky, Dungeon, and Rainbow are the least common Slimes (hence the bold)

You can use a Clentaminator with solution to change biomes and make slimes appear. It is also handy to use a Battle Potion or hold a Water Candle to make slimes spawn more frequently. It is also possible to idle boost this by making preparations, such as a level surface as shown by a video in the comments from Fire Hawk GYJS.

***Same as 360 version***

Credit to Imaginaryy for his original list for 360 version.

I am writing this guide specifically for those I have carried through the game. The solution will make use of the items provided and the seed used for the achievements.
For anyone else wanting to follow along, the following seed will be required:

Size - Small
Evil - Corruption
Seed - 485539397

I would also highly suggest playing on the Journey difficulty as this mode allows you to manipulate the spawn rate (this should be set to 10), time of day and weather conditions, which saves a lot of time.
You should also freeze time when starting and set it to noon. There are less enemy types found during the day which increases the odds of Slimes spawning.

For those I haven't helped, this guide assumes you have access to a Terraformer / Clentaminator and the appropriate solutions to create Crimson, Corruption and Hallow.
You will also need a GPS device to view the coordinates.

Before starting the hunt for the slimes, you will need to make sure that The Wall of Flesh and Skeletron have been defeated to gain access to Hard-Mode and the Dungeon respectively. For those I have helped, this has already been done for you.

Finally, it's worth mentioning that you will likely have come across and killed some of these Slimes whilst you were receiving help or working on other achievements or Slime kills. As such, you should consult your Bestiary before starting, in any worlds you have played in, to see which Slimes have already been killed. Although it's worth noting, that in some cases, Slimes killed by other players won't count for you. As such it may be worth killing each slime again just to be certain.

The following Slimes are easier to find in pre hard-mode as there are less potential enemies that can spawn:

1. Green Slime - Found at the spawn point (Centre, 138' Surface)
2. Blue Slime - Found near the right-side ocean (3444' East, 112 Surface)
3. Red Slime - Found in the Underground (784' East, 199' Underground)
4. Purple Slime - Will appear in aboveground Forests, but not near the spawn (3069' West, 221' Surface)
5. Yellow Slime - Found in the Underground (784' East, 199' Underground)
6. Black Slime - A somewhat rare spawn in the Cavern Layer (3458' East, 1090' Caverns)
7. Mother Slime - Found in the Caverns (798' East, 624' Caverns)
8. Baby Slime - Found in the Caverns and will appear when a Mother Slime is killed (798' East, 624' Caverns)
9. Pinky - A Very Rare spawn that can appear in any location. If you don't come across this Slime by the time you have killed everything else you should start a brand new world and go AFK in a Forest (but not the spawn location) with the Stardust Armour equipped and the spawn rate set to 10x. Hopefully, by the time you come back, the golem will have killed a Pinky for you.
10. Jungle Slime - Found in the Surface Jungle (2071' West, 196' Surface)
11. Spiked Jungle Slime - Found in the Underground Jungle (2241' West, 400 Caverns)
12. Lava Slime - Use the Shell Phone to teleport to the Underworld or dig down (177' East, 1434' Underworld)
13. Dungeon Slime - Found in the dungeon (3060' East, 986' Caverns)
14. Ice Slime - Found in the snow biome on the right side of the map (1707' East, 125' Surface)
15. Umbrella Slime - Can be found aboveground whenever it is raining (1707' East, 125' Surface)
16. Spiked Ice Slime - Found in the underground Snow Biome (1408' East, 896' Caverns)
17. King Slime - Can appear in a number of different ways, but more easily summoned using a Slime Crown (Crafted at an alter using 20 Gel and a Gold / Platinum Crown [5 Gold / Platinum Bars and one Ruby]). If I have carried you through the game, this boss will have already been killed.

These Slimes are found in specific evils and mostly in Hard-Mode. If you have the Terraformer, you can use it to create the appropriate biomes by swapping the solutions in the ammo slot of your inventory. This allows you to get multiple Slimes in the same location.

1. Crimslime - Found in the Underground Crimson, this can be created anywhere using the Terraformer (784' East, 199' Underground) This slime is incredibly buggy and may take dozens of kills before finally registering. This is likely due to this particular slime having three size variants that all need to be killed before the slime is properly tracked
2. Corrupt Slime - Found in the Corruption, this can be created anywhere using the Terraformer (784' East, 199' Underground)
3. Slimeling - Found in the Underground Corruption and will appear after a Corrupt Slime has been killed, this can be created anywhere using the Terraformer (784' East, 199' Underground)
4. Slimer - Found in the Corruption, this can be created anywhere using the Terraformer (784' East, 199' Underground)
5. Toxic Sludge - Found in the Underground (784' East, 199' Underground)
6. Illuminant Slime - Found in the underground Hallow (784' East, 199' Caverns)
7. Rainbow Slime - Found in the Hallow when raining (Centre, 138' Surface)

You can consult the Bestiary found in the menu to see which Slimes have been killed in your current world. Using the filter feature, you can do a manual search for "Slime" which will only show the Slimes that have been killed. This doesn't include Pinky who will need to be searched for separately.
I would also suggest tracking the achievement as it can sometimes take multiple kills of the same Slimes before it registers for the achievement.
As a final note, you can change the ammo the Terraformer / Clentaminator uses by moving the ammo you want to fire into the topmost ammo slot.

-GREEN SLIME (bioma da floresta)
-BLUE SLIME (bioma da floresta)
-RED SLIME (subsolo)
-PURPLE SLIME (pode aparecer nas camadas externas de qualquer bioma)
-YELLOW SLIME (subsolo)
-BLACK SLIME (cavernas)
-MOTHER SLIME (cavernas)
-BABY SLIME (surge ao matar a MOTHER SLIME)
-PINKY (muito raro/bioma da floresta)
-LAVA SLIME (submundo)
-DUNGEON SLIME (The dungeon)
-ICE SLIME (bioma de gelo)
-SPIKED ICE SLIME (subsolo do bioma de gelo)
-JUNGLE SLIME (selva)
-SPIKED JUNGLE SLIME (subsolo do bioma da selva)
-UMBRELLA SLIME (pode surgir em qualquer bioma, ENQUANTO ESTIVER CHOVENDO)
-KING SLIME (existe uma chance de ele aparecer no lugar de qualquer slime na superfície/ela aparece após matar 150 slimes durante o evento SLIME RAIN (75 caso já tenha aparecido alguma vez) /pode ser invocado com o item SLIME CROWN)
-CORRUPT SLIME (durante o hardmode/bioma da corrupção)
-SLIMELING (durante o hardmode/surge ao matar o CORRUPT SLIME)
-SLIMER (surge durante o hardmode/bioma da corrupção (voando))
-CRIMSLIME (durante o hardmode/surge no bioma carmesin)
-TOXIC SLUDGE (durante o hardmode/subsolo/pode surgir no lugar de outros slimes da área)
-ILLUMINANT SLIME (durante o hardmode/surge nas camadas do subsolo e cavernas do bioma consagrado (HALLOW))
-RAINBOW SLIME (muito raro/durante o hardmode/pode surgir na superfície do bioma consagrado (HALLOW), durante a chuva)
*ATENÇÃO: não confundir camadas (superfície, subsolo, cavernas, etc) com biomas (floresta, selva, gelo, corrupto, etc)