Medivh spent a happy childhood in his father's court and developed great friendships with Prince Llane Wrynn, the future king of Stormwind, and the future warrior Anduin Lothar. When Medivh turned fourteen, Aegwynn's powers awakened in him, killing his father and seriously injuring Medivh. He lay in a coma at Northshire Abbey for six years while Sargeras took control of him.
Medivh awoke at age 20. Under the direction of Sargeras, he began to seek out a way to destroy the humans of Azeroth. He looked to the distant world of Draenor (hey, that's where we all are now!) and made a bargain with the orc warlock Gul'dan. Medivh told Gul'dan the location of Sargeras's Tomb in exchange for Gul'dan's agreement to invade Azeroth with the Horde. (No, not the modern, peacenik Thrall-and-tauren Horde, the old nasty Orc one.) Medivh opened the Dark Portal to Outland, allowing the Horde to come through and fight the humans. This was known as the First War.
Medivh didn't proclaim his part in the war, but his new apprentice, Khadgar, grew suspicious. Aegwynn tried to reason with Medivh and finally fought him. Medivh won and banished Aegwynn from his sight. But the battle had weakened him, and he drew strength from the land of Azeroth itself. Khadgar finally figured out what Medivh was doing and went to his childhood friends Llane and Lothar to ask for their help. Along with the Horde emissary Garona Halforcen, Lother and Khadgar led human troops to fight Medivh. In the ensuing battle, Khadgar drove a sword through Medivh's heart, killing him. But we all know that no one ever really dies in Warcraft ...
A long, long time passed, and the Burning Legion was once again poised to invade Azeroth. A mysterious prophet appeared to the young warchief of the Horde, Thrall, and told him to go to the western land of Kalimdor. Thrall took the prophet at his word and headed over. The prophet then tried to convince the humans to do the same, but most of them ignored him. Only Jaina Proudmoore sent forces to Kalimdor. Thrall and Jaina eventually met up, but as they were ready to fight each other, the prophet told them to stop fighting and ally to destroy the Burning Legion. They reluctantly agreed. Finally, in a dream vision to Thrall and the night elf Malfurion Stormrage, the prophet revealed that he was in fact Medivh, trying to make up for helping Sargeras. The Horde and the Alliance worked together for once and destroyed the demonic Archimonde. His job complete, Medivh vanished to "take his place among the legends of the past."
Where he is now: Among the legends of the past, wherever that is. Medivh has a strong influence on many things in World of Warcraft. Atiesh, Greatstaff of the Guardian, is the staff that Medivh used in life. Khadgar, who killed Medivh, stands in the middle of Shattrath City. The entire instance of Karazhan is set in Medivh's former tower, and rumor has it that an "echo" of Medivh is seen in the instance. In the second Caverns of Time instance, you have to preserve Medivh's opening of the Dark Portal for some really complicated reasons that basically amount to "Don't kill your own grandmother!" Medivh is still technically dead, but it's definitely possible for his spirit to return in the guise of the Prophet if Azeroth is threatened once more.
Identifying characteristics: As the Prophet, Medivh can transform into a raven. His human form has a beard, and he's generally dressed in robes. Like most mages, his hands are often glowy.
Why we need to
kill him clean out his tower: Khadgar and Lothar apparently didn't bother to clean out Medivh's tower after killing him, so it's kind of taken on a life of its own. Medivh's servant lives there, and ghosts still protect their former master's property. Prince Malchezzar, an eredar who serves the Burning Legion, currently rules over the tower. Basically, you're the sheriff come to evict a really annoying tenant, except in this case half the residents are ghosts and the tower is trying to kill you.
For more information: Medivh on WoWWiki, Medivh on Wikipedia, Medivh opening the Dark Portal in Caverns of Time.
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(Edited for Tirasfal/Tirisfal spelling mistake.)