What was the pratigna (oath) and Why Bhishma took pratigna in Mahabharata? And to whom?
The Oath Bhishma taken was Brahmacharya (study and meditation in celibacy. He took it for the benefit of his father Santanu.
The oath he took taken is explained in chapter 100 of Adi Parva of Mahabharata.
Once, King Santanu went for hunting and was attracted to a fragrance coming from an unknown direction. He follows that fragrance and finds a beautiful woman, the daughter of Fisherman chief. He desired her and holding her hand, he wanted to marry her. He asks the consent from her father but her father asks Monarch Santanu :
O king, as soon as my daughter of superior complexion was born, it was of course, understood that she should be bestowed upon a husband. But listen to the desire I have cherished all along in my heart. O sinless one, thou art truthful: if thou desirest to obtain this maiden as a gift from me, give, me then this pledge. If, indeed, thou givest the pledge, I will of course bestow my daughter upon thee for truly I can never obtain a husband for her equal to thee.
Santanu asks what is his desire and the fisherman chief puts a condition,
'O king, what I ask of thee is this: the son born of this maiden shall be installed by thee on thy throne and none else shall thou make thy successor.'
Hearing this, Santanu was shocked and returns to his kingdom sorrowfully. Devavrata who has been observing his father asked the reason for his sorrow. Santanu doesn't answer him properly and doesn't narrate the incident. Intelligent Devavrata notices something is wrong and goes to an old minister and asks the reason of King's grief. When asked thus by the prince Devavrata, old minister tells him about the boon the Fisherman chief demanded. For the benefit of his father, he goes to that place along with the best Kshatriya chiefs and asked him his daughter on behalf of the king. The chief of the fishermen received him with due adorations and addressed to him
O bull among the Bharatas, thou art the first of all wielders of weapons and the only son of Santanu. Thy power is great. But I have something to tell thee. If the bride's father was Indra himself, even then he would have to repent of rejecting such an exceedingly honourable and desirable proposal of marriage. The great man of whose seed this celebrated maiden named Satyavati was born, is, indeed, equal to you in virtue. He hath spoken to me on many occasions of the virtues of thy father and told me that, the king alone is worthy of (marrying) Satyavati. Let me tell you that I have even rejected the solicitations of that best of Brahmarshis--the celestial sage Asita--who, too, had often asked for Satyavati's hand in marriage. I have only one word to say on the part of this maiden. In the matter of the proposed marriage there is one great objection founded on the fact of a rival in the person of a co-wife's son. O oppressor of all foes, he hath no security, even if he be an Asura or a Gandharva, who hath a rival in thee. There is this only objection to the proposed marriage, and nothing else. Blest be thou! But this is all I have to say in the matter of the bestowal or otherwise, of Satyavati.
Devavrata having heard these words and moved by the desire of benefiting his father, uttered these words.
'O foremost of truthful men, listen to the vow I utter! The man has not been or will not be born, who will have the courage to take such a vow! I shall accomplish all that thou demandest! The son that may be born of this maiden shall be our king.'
Then the fisherman puts his another doubt.
Thus addressed, the chief of the fishermen, impelled by desire of sovereignty (for his daughter's son), to achieve the almost impossible, then said, 'O thou of virtuous soul, thou art come hither as full agent on behalf of thy father Santanu of immeasurable glory; be thou also the sole manager on my behalf in the matter of the bestowal of this my daughter. But, O amiable one, there is something else to be said, something else to be reflected upon by thee. O suppressor of foes, those that have daughters, from the very nature of their obligations, must say what I say. O thou that art devoted to truth, the promise thou hast given in the presence of these chiefs for the benefit of Satyavati, hath, indeed, been worthy of thee. O thou of mighty arms, I have not the least doubt of its ever being violated by thee. But I have my doubts in respect of the children thou mayst beget.
Having heard the words uttered by the fisherman, Devavrata takes another terrible vow. It is this vow for which he got the name Bhishma.
Chief of fishermen, thou best of men, listen to what I say in the presence of these assembled kings. Ye kings, I have already relinquished my right to the throne, I shall now settle the matter of my children. O fisherman, from this day I adopt the vow of Brahmacharya (study and meditation in celibacy). If I die sonless, I shall yet attain to regions of perennial bliss in heaven!
"Vaisampayana continued, 'Upon these words of the son of Ganga, the hair on the fisherman's body stood on end from glee, and he replied, 'I bestow my daughter!' Immediately after, the Apsaras and the gods with diverse tribes of Rishis began to rain down flowers from the firmament upon the head of Devavrata and exclaimed, 'This one is Bhishma (the terrible).' Bhishma then, to serve his father, addressed the illustrious damsel and said, 'O mother, ascend this chariot, and let us go unto our house.'
Due to his Bhishma pratigna(terrible oath), he got his popular name Bhishma which was Devavrata originally.