high school vs Gymnasium

high school and Gymnasium both are nouns.

Word	Noun	Adjective	Verb	Adverb
high school	Yes	No	No	No
Gymnasium	Yes	No	No	No

As nouns, Gymnasium is a hypernym of high school; that is, Gymnasium is a word with a broader meaning than high school:

high school: a public secondary school usually including grades 9 through 12
Gymnasium: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of high school include lycee, lyceum, middle school, secondary school.

high school (noun)	Gymnasium (noun)
a public secondary school usually including grades 9 through 12	a school for students intermediate between elementary school and college; usually grades 9 to 12