The original Enemy was the dark ValaMelkor. After his murder of Finwë and theft of the Silmarils, he was named Morgoth, 'Black Enemy', by Finwë's son Fëanor, and was known by that name ever afterwards.
Sauron was a servant of the first Enemy during the Elder Days, but after the fall of his master Morgoth, he took on the mantle of Dark Lord and Enemy of Elves and Men. The Free Peoples were reluctant to speak his true name, and thus he was often referred to simply as 'the Enemy'.