home-fried potatoes vs tater

home-fried potatoes and tater both are nouns.

Word	Noun	Adjective	Verb	Adverb
home-fried potatoes	Yes	No	No	No
tater	Yes	No	No	No

As nouns, tater is a hypernym of home-fried potatoes; that is, tater is a word with a broader meaning than home-fried potatoes:

home-fried potatoes: sliced pieces of potato fried in a pan until brown and crisp
tater: an edible tuber native to South America; a staple food of Ireland

Other hypernyms of home-fried potatoes include Irish potato, murphy, potato, spud, white potato.

home-fried potatoes (noun)	tater (noun)
sliced pieces of potato fried in a pan until brown and crisp	an edible tuber native to South America; a staple food of Ireland