Walkthrough
When nearing the Tristram Cathedral the player comes across a group of zombies bent over something. After killing them, there is an Injured Villager laying on the ground. With his last breath he tells you the story of King Leoric. The quest description says: "Descend into the catacombs and destroy the arisen Skeleton King and his army of the undead".
Ghost of Leoric starting up the pillars.
Following this event and going deeper into the cathedral's catacombs killing monsters, the player will find a room with four huge pillars. Reaching approximately the middle of the room, a ghost of King Leoric appears and say: "You dare bring the warmth of life into my tomb!? Prepare yourself mortal to serve me for all eternity!" The king raises his hands, starting to glow, followed by the four pillars sprouting blue from the tops.
Skeletons start to appear from everywhere, summoned by the pillars, and the player must destroy the pillars or face being overwhelmed by skeletons of different sorts.
A bit further along in the Cathedral, the player finally finds King Leoric's throne room. When the player gives the crown back on the king, his ghost is summoned to the bones and he is reanimated. He laughs menacingly and Skeletons start to emerge from the sides. The player now has to keep away from skeletons and King Leorics huge mace as well as damaging and killing the boss.
Reward
The quest in the full game will probably give some nice reward, but in the demo, players were rewarded with the message that they had completed the BlizzCon demo.
Background
This quest is reminiscent of the the Butcher quest in Diablo I; both are given to the player by a dying villager encountered just outside the Tristram Cathedral.
That little Easter Egg might have been just for the Blizzcon demo; it's unknown how the quest will be presented in the final game.
Related NPCs
Video
Leoric's reanimation and the subsequent battle can be seen from just after the Crown was used to awaken him, in the following YouTube video.