The Prophetess of Apollo is the lady who works at the temple of Apollo in Delphi. Specifically, she is the person that mortals talk to if they have any questions to ask the god. After a mortal asks his or her question, the Prophetess channels the divine power to provide an answer. She's like a divine Google.
In the opening scene of the play, the Prophetess comes onstage and talks about the long and complicated back-story of which gods inhabited the oracle at Delphi, ending with Apollo. Then she is the one who discovers Orestes and the Furies sleeping inside the temple:
(Prophetess): "In front of this man an amazing band is asleep, of women, sitting on the chairs—no, I do not mean women, but Gorgons […] these however have no wings to be seen; and they are black, utterly revolting in their manner, snoring out a breath which is unapproachable […] (46-59)
After this brief role—giving us a little history lesson and letting us know just how pukesome the Furies are— she wisely hightails it offstage.