academy vs secondary school

academy and secondary school both are nouns.

Word	Noun	Adjective	Verb	Adverb
academy	Yes	No	No	No
secondary school	Yes	No	No	No

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

academy: a secondary school (usually private)
secondary school: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of academy include Gymnasium, lycee, lyceum, middle school.

academy (noun)	secondary school (noun)
a learned establishment for the advancement of knowledge	a school for students intermediate between elementary school and college; usually grades 9 to 12
a secondary school (usually private)
a school for special training
an institution for the advancement of art or science or literature