The Elder Slime is a town pet. The Elder Slime does not count towards NPCs' "crowded" limit and hence will not affect NPC happiness, although it does count as a regular town NPC for the purpose of the Pylon network. The Elder Slime spawns randomly in open areas in the Cavern layer after defeating Skeletron and will spawn as an Old Shaking Chest that can be opened with a Golden Key.
Elder Slime in Suitable Housing
Elder Slime after being released from the Old Shaking Chest