Cassandra is a princess of Troy. We aren't really told much about her back-story in Aeschylus's Agamemnon, but other ancient sources report that she was captured by the Greek warrior Ajax, who pried her loose from an altar where she had claimed sanctuary. In any case, she ended up being Agamemnon's captive; Agamemnon says that she "has followed with me as the army's gift, the flower chosen out of many things" (953-955). Aeschylus's original audience could have seen in this a reference to Homer's Iliad, in which Agamemnon is forced to give up two women he had taken prisoner: Chryseis, the daughter of the priest of Apollo, and then Briseis, the girlfriend of Achilles. If the Greek soldiers gave Agamemnon Cassandra, this could be seen as payback for the two previous women he had to give up.
Be that as it may, Cassandra is much more than merely a prisoner. Even though she's quiet at first, she has a story and strong identity of her own; once she starts speaking, the Chorus and the readers aren't likely to forget it. The basics are these: as Cassandra reveals, the god Apollo once developed a crush on her, and tried to have an affair with her, as he did with many other mortal women. The details of what happened are a little unclear, but, basically, Apollo gave her the gift of prophecy at some early stage in their "relationship." Then, Cassandra says she "cheated" the god, though she doesn't specify what this means. Whatever happened, Apollo now modified his initial blessing by throwing a curse on top of it. Even though Cassandra would be able to tell the future (and look into the past), she was also condemned never to be believed.
We can see how this plays out in Cassandra's big scene with the Chorus, in which she uses her gift to reveal the murder that took place in the house of Agamemnon, and predict her own imminent death, as well as Agamemnon's, at the hands of a woman. Even though the Chorus says they believe her, they don't do anything concrete to help her. (Maybe what Apollo's curse really means is that Cassandra will be able to prophecy but without it making any actual difference.)
What is interesting here, however, is how Cassandra's attitude changes from one of total horror to one of acceptance. She accepts that there is no way out, and is also contented by the thought that, with Agamemnon's death, the sufferings of her fellow Trojans will be avenged. She is also contented by the thought that someone will come and avenge Agamemnon by killing Clytemnestra; she prays that, whoever that person may be, he will avenge her too. Interestingly enough, however, Cassandra always refers to her destroyer as Apollo; thus, she sees herself as merely passing through the drama of Agamemnon and Clytemnestra – what matters for her is her unfinished business with the god Apollo.