My grandmother saw both billie holiday and ella fitzgerald perform. what kind of conjunction are the underlined words?

I'm assuming the underlined conjunctions are both and and (because you didn't underline any, and those are the only two conjunctions in the sentence).
Those conjunctions are called correlative conjunctions - this means that they are connecting two items which are equally important. So, this woman saw both Billie Holiday and Ella Fitzgerald - and both of those artists are important to the woman.