The Cursing of Melkor

Morgoth is the chief title of Melkor, He Who Arises in Might, formerly the greatest of the Valar, those of the Ainur who came to govern Arda. Melkor repeatedly fought against the Valar to corrupt and destroy that which they created, and was once bound in Mandos for three ages for his crimes. Upon his release from this imprisonment, he deceived Manwë, the King of Arda (for Manwe could not comprehend his evil) and walked among the Valar and the Elves in the society of Aman. He grew in friendship with all the Noldor and taught them great crafts, and all learned from him save Fëanor, Prince of the Noldor and son of Finwë their king.

Feanor was the creator of the Silmarils, which are Jewels that contain the light of the Two Trees of Valinor. These he created without help or knowledge of Melkor, and Melkor coveted them. Melkor therefore sowed strife among the houses of the Noldor in order that he might turn the Noldor against the Valar and steal the Silmarils for himself. He succeeded in angering Feanor toward his brother Fingolfin, but when both were summoned to the council of the Valar, the malice of Melkor was laid bare, and he fled from the pursuit of Tulkas and Oromë of the Valar.

Melkor fled to the south of Aman and entered into league with Ungoliant, a creature of great evil, who feared light and lusted after it. Together they covered themselves with a great darkness and went to Valimar at a time of festival. Then they approached the Two Trees, and Melkor wounded them, and Ungoliant drank their sap and grew giant and terrible as she consumed the Light. Then they fled from Valimar to the north, Melkor's purpose achieved.

And then the Valar summoned all to Ezellohar, the Green Mound where the Trees had stood, and Yavanna declared that she could not restore them save she should open the Silmarils, for they alone contained their light still. But Fëanor denied her, for the lies of Melkor lived in his heart, and he feared the Valar as thieves of his works. For Melkor had said to him, that the Silmarils were not safe as long as the Valar would possess them.

But at that moment a rider came from Formenos, the stronghold of Fëanor and Finwë, saying that the King was slain, and Melkor had despoiled the place and taken the Silmarils. And Feanor was wroth, and cursed Melkor, naming him Morgoth, which is Black Foe of the World -- and by that name was he known ever after.

The Doings of Morgoth

Morgoth became the Lord of Angband and the Great Enemy of the Elves in Middle-Earth (both the grey-elves, the Sindar, and the deep-elves, the Noldor.) Many were his evil works and great the destruction he wrought in Beleriand until at last Eärendil the Blessed sailed to Aman and sought the pardon of the Valar on behalf of both Elves and Men, for he was of both of the Two Kindreds. Then the Valar returned to Middle-earth, and assailed Morgoth, and there was fought the War of Wrath, of which few tales tell. And then the Valar took Morgoth, and bound him again with the chain Angainor, and put him out of Ea and into the Void, where he remains bound.

The Nature of Morgoth

Morgoth was originally one of the Ainur, the ones who made the Great Music. Iluvatar is the "God" of the Tolkien mythos, and while the Valar (Ainur who descended into Arda) are referred to as "gods" they are more properly called the servants of Iluvatar and the Lords of Arda. Morgoth was Melkor, the greatest of the Ainur, who rebelled against Iluvatar out of pride, and sought alone for the Flame Imperishable, but he "found not the Fire, for it is with Iluvatar."

Morgoth is definitely the "Satan" of Tolkien's mythos. It was his lies that caused the "Fall" of the Elves (the Oath of Fëanor and the Kinslaying) and The Silmarillion also speaks of him leaving his stronghold at Angband to travel east and corrupt the first of Men. Beor the Old spoke of this to Finrod Felagund when he said,

"A darkness lies behind us, and we turn our backs upon it, and do not desire to return thither even in thought." (The Silmarillion, pp. 141)

It is important to remember that while Tolkien did not write allegories in that sense, he often said he intended to tell the history of the Elves a world that could be our own (See The Silmarillion pp. xiv.) His mythology was designed that a Christian man could still believe that these things had actually happened once, long ago. In that sense, Morgoth, as the Great Enemy, is "Satan". Tolkien's writings are not allegorical, they are a history that if it were true, would be intimately linked with our own - including sharing the same Satan. If you were to actually believe Tokien's fictional tales as history, then Morgoth would be the being that we know as Satan today (assuming you even believe in Satan.)

Disclaimer: I definitely do not believe Tolkien's writings ever actually happened. They are a work of fiction. However, one can gain much insight into Tolkien's intentions from treating them as an actual history, which I do here.