backgammon vs board game

backgammon and board game both are nouns.

Word	Noun	Adjective	Verb	Adverb
backgammon	Yes	No	No	No
board game	Yes	No	No	No

As nouns, board game is a hypernym of backgammon; that is, board game is a word with a broader meaning than backgammon:

backgammon: a board game for two players; pieces move according to throws of the dice
board game: a game played on a specially designed board

backgammon (noun)	board game (noun)
a board game for two players; pieces move according to throws of the dice	a game played on a specially designed board