The Slime King is a boss that randomly spawns. When it spawns near the player, a health bar similar to the one when fighting the Ender Dragon will appear at the top of the HUD. It also seem to spawn various versions (read Infernal Mobs) of Blue Slimes as minions.
Combat Strategy
The Slime King has a health of 100 hitpoints, and a decent attack as well, but its movement is rather slow. The easiest way to deal with it is either with jumping forward for a hit and moving back, or to take it on from a distance with a bow.
Once you have defeated it, you will recieve a "King Slime" tool; a King Slime Hatchet.