Slimes are everywhere in Terraria, and there are many varieties. Most all slimes will be encountered through natural progression of the game, so you should never worry about hunting for a specific one save for Pinky if you're unlucky.

Green Slime Spawn in Forests.
Blue Slime Found in the Forest Underground.
Red Slime Found in the Underground layer.
Purple Slime Found in Forests.
Yellow Slime Found in the Underground layer.
Black Slime Found in the Cavern layer.
Mother Slime Black-colored and larger. Found in the Cavern layer. Spawns 2 to 3 Baby Slimes on death.
Baby Slime Spawns when a Mother Slime is killed. The same color as a Mother Slime, but smaller.
Pinky A rare slime that spawns in Forests and the Underground layer. Smaller and tougher than other slimes listed above.
Jungle Slime Found in the Jungle. Green-colored.
Lava Slime Glowing orange slime found in the Underworld.
Dungeon Slime Somewhat uncommon indigo-colored slime found in the Dungeon.
King Slime A very large blue-colored slime. Considered a boss. It can spawn very rarely in Forests that are not in the center third of the map (so basically they can't spawn around your spawn point). It's easier to just summon one by using a Slime Crown, crafted from a Gold Crown and 99 Gel at a Demon Altar.
Corrupt Slime Found only in Hardmode in the Corruption and Underground Corruption. Dark blue-colored and spawns three Slimelings upon death.
Slimeling Found only in Hardmode. Spawns when a Corrupt Slime is killed. Same color as a Corrupt Slime, only smaller.
Shadow Slime Found only in Hardmode in the Corruption and Underground Corruption. Same color as the Corrupt Slime, but smaller. Console-exclusive.
Slimer A winged flying slime found only in Hardmode in the Corruption and Underground Corruption. Dark blue-colored. Has two "forms": flight form and normal. You first kill its wings, then you can kill the slime completely.
Toxic Sludge Found only in Hardmode in the Underground layer. Green-colored with a happy face.
Illuminant Slime Found only in Hardmode in the Underground Hallow. Bright glowing purple-colored slime.