“Elf” is an uncommon word.  The hyphenated combination is unattested in OED, therefore a JRRT original.  Tolkien spun out a marvelously complex tapestry of interrelations between types of Elves in his legendarium.

[08.131]  The feasting people were Wood-elves, of course.  These are not wicked folk.  If they have a fault it is distrust of strangers.  Though their magic was strong, even in those days they were wary. They differed from the High Elves of the West, and were more dangerous and less wise. For most of them (together with their scattered relations in the hills and mountains) were descended from the ancient tribes that never went to Faerie in the West. There the Light-elves and the Deep-elves and the Sea-elves went and lived for ages, and grew fairer and wiser and more learned, and invented their magic and their cunning craft in the making of beautiful xx and marvellous things, before some came back into the Wide World. In the Wide World the Wood-elves lingered in the twilight of our Sun and Moon…

• 08.131 and the Deep-elves

The Tolkien Gateway article on them may be found here.  A beautifully detailed genealogy of individuals may be found here.  Finally, a very accessible chart of the relationships between different groups of Elves may be found here.