dancer vs performer

dancer and performer both are nouns.

Word	Noun	Adjective	Verb	Adverb
dancer	Yes	No	No	No
performer	Yes	No	No	No

As nouns, performer is a hypernym of dancer; that is, performer is a word with a broader meaning than dancer:

dancer: a performer who dances professionally
performer: an entertainer who performs a dramatic or musical work for an audience

Other hypernyms of dancer include performing artist.

dancer (noun)	performer (noun)
a performer who dances professionally	an entertainer who performs a dramatic or musical work for an audience
a person who participates in a social gathering arranged for dancing (as a ball)