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
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.