Both names refer to the same person, butperson; Melkor is the properfirst name he was created withknown by, while Morgoth is a name given to him by his enemies.
Melkor, the original name, is the one he had from the very beginning, when he was part of the music of Iluvatar. I don't know if it has a meaning inThe name is Quenya or in some Valarin speech, but it's his "real" nameorigin, and usedmeans "he who arises in the earliest storiesmight", up untilas he was accounted the destructionmightiest of the Ainur who entered into Arda (and were thus called the Valar). He was called by this name until the Two Trees in Valinor died by his hand (with the aid of the spider creature Ungoliant), the Silmarilli were stolen and he fled from Valinor to Middle-earth.
After that momentuousmomentous event, he was given the name Morgoth (or, fully, Morgoth Bauglir) by Feanor. Morgoth is in Sindarin, and means "Dark Foe" or some similar variant (You can see the root "Mor", for dark, repeated in many names, such as Mordor. "Goth" can be found in Gothmog, lord of the Balrogs). "Bauglir", similarly, means "Tyrant" or "Oppressor".
The Feanor rose, and lifting up his hand before Manwe he cursed Melkor, naming him Morgoth, the Black Foe of the World; and by that name only was he known to the Eldar ever after.
The Silmarillion, ch. 9.
So properly, Melkor is the only true name he has. "Morgoth" is an appelation given to him by his enemies, declaring him their foe to the end of time.