This is discussed in the Silmarillion. In short, Melkor is his proper name (from the Quenya word meaning "One who arises in Might") but the Elves won't say it for, ahem, reasons.
Of the Enemies Last of all is set the name of Melkor, He who arises in Might. But that name he has forfeited; and the Noldor, who among the Elves suffered most from his malice, will not utter it, and they name him Morgoth, the Dark Enemy of the World. Great might was given to him by Ilúvatar, and he was coeval with Manwë. - Silmarillion - Of the Flight of the pNoldor"Noldor
and from the index of names at the rear of the book
Melkor: The Quenya name for the great rebellious Vala, the beginning of evil, in his origin the mightiest of the Ainur; afterwards named Morgoth, Bauglir, the Dark Lord, the Enemy, etc. The meaning of Melkor was 'He who arises in Might'; the Sindarin form was Belegur, but it was never used, save in a deliberately altered form Belegurth 'Great Death'. Passim (after the rape of the Silmarils usually called Morgoth) Silmarillion - Index of Names
and from his Letter 211
In the cosmogonic myth Manwe is said to be ‘brother’ of Melkor, that is they were coeval and equipotent in the mind of the Creator. Melkor became the rebel, and the Diabolos of these tales, who disputed the kingdom of Arda with Manwë. (He was usually called Morgoth in Grey-elven.)