Kill one of each slime listed below :
Slime Type OR Parent Type/Child Type - Biome
Pre-Hardmode
Green - Forest (Day)
Blue - Forest (Day), Underground
Purple - Forest (Day)
Umbrella - Rain (Day)
Pinky - Forest, Underground, Cavern
Red - Underground
Yellow - Underground
Black - Cavern
Mother/Baby - Cavern
Dungeon - Dungeon
Ice - Snow
Spiked Ice - Ice (Snow Caverns)
Jungle - Jungle
Spiked Jungle - Underground Jungle
Lava - Hell
King Slime - Kill 150 slimes in Slime Rain (75 if its been defeated at least once before), use a Slime Crown, or find it in the outer sixths of the map where it spawns rarely during the day.
Hardmode
Toxic Sludge - Underground (replaces green slimes)
Corrupt/Slimeling - Corruption, Underground Corruption
Slimer - Corruption, Underground Corruption
Crimslime - Crimson
Illuminant - Underground Hallow
Rainbow - Hallow (Rain)