Find and kill each of the following Slimes to pop the achievement. Some will appear only in certain areas, and some are only present in hard mode.
SPECIAL NOTE from Daeryoon: 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!
1 - Green SlimeCommon near surface.
2 - Blue SlimeCommon near surface.
3 - Red SlimeCommon near surface.
4 - Purple SlimeCommon near surface.
5 - Yellow SlimeCommon near surface.
6 - Black Slime Found underground.
7 - Mother SlimeFound underground.
8 - Baby SlimeFound underground. Spawns from Mother Slimes.
9 - PinkyFound anywhere; rare spawn rate.
Suggested: Create new small worlds, search surface, and go to underground stone layer just before the underworld. Should assist in locating a pinky slime. (credit to Crimson Drifter & sonnyforple)
**more info in comments
10 - Jungle SlimeFound in the jungle biome.
11 - Lava SlimeFound in the underworld.
12 - Dungeon Slime Found in the dungeon.
13 - King Slime Regular boss; summoned with Slime Crown; or rare spawn rate.
14 - * Corrupt Slime * HMFound in the corruption.
15 - * Slimeling *HMFound in the corruption. Spawns from Corrupt Slimes.
16 - * Slimer *HMFound in the corruption.
17 - * Toxic Sludge *HMFound underground.
18 - * Illuminant Slime *HMFound in the hallowed area, or underground hallowed (CR: Dang3r Gaming)
19 - * Shadow Slime *HMFound in Corruption. Looks similar to Corrupt Slime.[1.2 update: slimes 20-25 added: credit to Daeryoon]
20 - Ice Slime
Found in Frozen biome.
21 - Frozen Spiked Slime
Found in Frozen biome. More common underground.
22. Spiked Jungle Slime
Found in Jungle biome. More common in Underground Jungle biome.
23. Umbrella Slime
Common spawn during Rain Event.
24. Rainbow Slime *HM
Rare spawn in hallowed area during Rain Event. **more info in comments
25. Crimslime *HM
Found in Crimson biome.
* HardMode only
The above is based on the wikia list, which worked for me,
with little tidbits added about where to find each slime.
Leave any comments, and if you vote down for any reason let me know why!
Original article(s) can be found here:http://terraria.gamepedia.com/Slime