Short answers, pulled from the story.
Cassandra was given the gift of true prophecy by Apollo but cursed so that no one would ever believe her predictions. The curse arose after she accepted Apollo's offer of foresight in exchange for becoming his lover, then refused to follow through on that promise. Since gods could not revoke a gift once given, Apollo ensured that every true prophecy she spoke would be dismissed as madness.
Apollo's curse ensured that every true prophecy Cassandra spoke would be dismissed. Her father Priam locked her in a chamber and kept her under guard, treating her as a madwoman. Even when she warned that Greek warriors were hiding inside the wooden horse, the Trojans stopped her, called her names, and refused to listen.
After Troy fell, King Agamemnon took Cassandra to Mycenae as his pallake, or concubine. She and Agamemnon were murdered by his wife Clytemnestra and her lover Aegisthus on their return. Cassandra, who had foreseen her own death, walked into her murder with full knowledge of what awaited her.
Ajax the Lesser was the Greek warrior who abducted and raped Cassandra in the temple of Athena during the sack of Troy, pulling the cult statue from its stand as he dragged her away. His sacrilege was punished by both Athena and Poseidon: Athena destroyed his ship with a thunderbolt, and when Ajax reached safety on a rock and boasted of surviving, Poseidon split the rock with his trident and cast him into the sea. His body washed ashore at Myconos, where Thetis buried him.
Various ancient sources record that Cassandra and Agamemnon had twin sons named Teledamus and Pelops. Both boys were murdered by Aegisthus.
Heinrich Schliemann, a German businessman and amateur archaeologist, claimed to have discovered the graves of Cassandra and Agamemnon in Grave Circle A at Mycenae, telegraphing the news to King George I of Greece. Later investigation showed the graves predated the Trojan War by at least 300 years, making the identification impossible.