eyrie vs bird nest

eyrie and bird nest both are nouns.

Word	Noun	Adjective	Verb	Adverb
eyrie	Yes	No	No	No
bird nest	Yes	No	No	No

As nouns, bird nest is a hypernym of eyrie; that is, bird nest is a word with a broader meaning than eyrie:

Other hypernyms of eyrie include bird's nest, birdnest.

eyrie (noun)	bird nest (noun)
any habitation at a high altitude	nest where birds lay their eggs and hatch their young
the lofty nest of a bird of prey (such as a hawk or eagle)