Melkor was the name of Morgoth before he rebelled against Iluvatar, much like Satan was Lucifer before he rebelled against God. Also much like Lucifer, Melkor was at the top of the heavenly hierarchy, given greater knowledge and power by Iluvatar than the other Valar (who were the highest of the Ainur, the heavenly beings. Valar were above the Maiar, who were also Ainur. Wizards were of the Maiar.)
But I digress.
As Iluvatar and the other Ainur sang Ea (Ea comprised the heavenlies and Arda, aka, Middle-earth) into existance, Melkor decided to sing his own songs, creating some of the more unpleasant things in Arda. He also had a following among the Maiar, the most loyal of these being Sauron, his head servant.
When Arda's creation was basically completed, some of the Ainur descended there to live and expand the creation. Melkor was one of them, but spent most of his time marring and distorting the creations of his heavenly brethren.
Melkor eventually established his base in northern Arda and named it Utumno. From his evil lair, Melkor accomplished an impressive resume of evil deeds, including the destruction and/or corruption of the two Lamps, the Spring of Arda, the Two Trees, and the theft of the Silmarils. He also bred all the monsters of Arda, including Balrogs, dragons, trolls, and orcs (Elves corrupted through torture). He spread enough lies, hatred, and other evils and poisons throughout Arda to corrupt it permanently (though not thoroughly).
Melkor was captured and cast into the Void during the Great Battle, before the Third Age. (I'm not sure exactly when.) Sauron continued the spreading of evil in Arda after his master's defeat.
Melkor's story is long and complicated, told mostly in the book The Silmarillion, where you can read the complete story of Melkor. There's really too much to say about the wars and destruction he was involved in to fit in one definition.