Read this post about an introduction to the Rasas.
The King of Kashi had three daughters, Amba, Ambika and Ambalika. He organized a swayamvara (a ceremony where the bride could choose her own groom from a group of assembled suitors) for his daughters.
On hearing about the swayamvara, Bhishma decided to go there and participate so that he could win the princesses for his half brother Vichitravirya. On reaching the venue, he publicly announced that he had chosen the princesses for his brother and that anybody who had an issue with that would have to face him in battle. Among other kings, Salva, who loved Amba challenged him in battle. However, Bhishma proved to be too strong for all challengers, defeated all of them and rode away with all three princesses.
After reaching Hastinapur, Amba approached him and told him that she and Salva were in love with each other and that she would be unable to marry Vichitravirya. She also told him as to how Salva was present at the swayamvara so that she could choose him as per the traditional custom and get married to him. Understanding her predicament and agreeing with her logic, Bhishma allowed Amba to go to Salva so that she could marry him. The other two sisters, Ambika and Ambalika however were married to Vichitravirya.
However when Amba reached Salva’s palace, he refused to marry her. His kshatriya ego was hurt as she was rightfully won in battle by Bhishma and now belonged to him. He refused to accept a woman who was another’s property. In fact, he went far enough to accuse Amba of happily leaving with her captor without feeling any remorse for his love. When Amba rebuked him and tried to convince him of her true love for him, he accused her of infidelity and rejected her outright.
Amba then had no choice but to go back to Hastinapur. Here, she was rejected by Vichitravirya who stated that he would not marry a woman whose heart belonged to somebody else. When Amba asked Bhishma to marry her, he reminded her of his vow to remain celibate forever. And thus, she was left alone and unwed without finding any suitable grooms.
Amba squarely placed the blame for her plight on Bhishma and swore to destroy him. She approached Parasurama and narrated her tale. She then requested him to defeat Bhishma and kill him in battle. Despite his valiant efforts, Parasurama failed in this endeavor of his.
As a last resort, Amba performed severe austerities and prayed to Lord Shiva, and when the Lord appeared before her, she asked for a boon – “Make me the cause of his death”. The Lord granted her wish, but on the condition that it would be so only in her next life.
To hasten her wish to come true, Amba then jumped into a funeral pyre to end this life of hers and soon be reborn in her next life to fulfill her destiny. She was then reborn as Shikhandi to King Drupada of Panchala who would ultimately be the cause of Bhishma’s death.
Thus, Amba’s raudram (anger, fury) towards Bhishma would end up being the principal cause of his death, despite the fact that he was granted a boon where his death would be of a time of his own choosing.
Image Courtesy: Devdutt Pattanaik’s illustration of Shikhandi and Arjuna in the battle