The fight between Parashurama, an incarnation of Vishnu, and Bhishma, the grandsire of Kauravas and Pandavas, is the battle of the equals. It was the battle between Guru and Shishya. Parashurama fought for getting justice for Amba, the eldest daughter of King of Kashi. Bhishma fought for defending the valor of Kshatriyas.
Amba, Ambika and Ambalika were daughters of King of Kashi. When the Swayamwara ceremony of the princesses was being held, Bhishma arrived there and defeated all the suitors and won the three maidens. Bhishma won the maidens through Viryashulka – a suitor who shows most valor wins the hand of maiden.
Bhishma performed this deed for Vichitravirya, the king of Hastinapura.
Just before the marriage ceremony, Amba informed Bhishma that she had already chosen the king of Shalva as her groom. So it would not be dharma for her to marry Vichitravirya as she has given her mind to another man.
Bhishma then allowed the eldest maiden to go to king of Shalva. But king of Shalva who was defeated by Bhishma in the Swayamvara ceremony did not accept Amba.
Amba then took shelter in an ashram. Here she met Parashurama, who along with other saints felt that Bhishma was responsible for the plight of Amba.
Parashurama challenged Bhishma in the Kuruskhetra to get justice to Amba. After taking the blessing of Parashurama, Bhishma mounted on his chariot. This act earned him the blessing of his preceptor.
They used all sort of weapons. Both were injured in the battle. Finally both of them used the Brahmastra. It created havoc in the sky. The heat of it was unbearable.
Then Sage Narada descended on earth and asked both Bhishma and Parashurama to stop the fighting.
Sage Narada reminded Parashurama that it was dharma of Kshatriya like Bhishma to fight and it is not possible to defeat a warrior who follows Dharma.
Bhishma also was reminded about the greatness of Parashurama and the reality that Parashurama can never be defeated.
Thus they both withdrew from the battlefield.