Your question was tagged with "chapter one," but there is no reference to Doctor Manette's story in chapter one of any of the three books which comprise A Tale of Two Cities by Charles Dickens. Though we have hints and suspicions throughout the book, we do not learn the complete story of Doctor Manette's imprisonment until book three, chapter ten.
Doctor Manette's son-in-law, Charles Darnay, is on trial for his life for the third time. He has tried to tell them he renounced his family title and holdings and has only come back to try to help his non-aristocrat friend; however, the rather bloodthirsty mob does not believe him and this time his own father-in-law has inadvertently become his accuser.
As evidence, Ernest Defarge produces a letter written by Doctor Manette fifteen years ago while he was a prisoner in the Bastille. Defarge found the letter in Manette's former cell, and now Defarge makes Manette read it, though of course Manette does not want to do so. The letter chronicles the events which lead to Manette's being falsely imprisoned in the Bastille more than a decade ago. This is his story.
Two evil twin brothers of the Evremonde family, were cruel and selfish men who cared for their own pleasures more than anything else, and there was little which could stop them from taking whatever it is they wanted. In this case, the thing they wanted was a young peasant girl, so they took her. In doing so, they caused her father to have a heart attack and die and demeaned her husband, who worked himself to death, just so they could have the thing they desired. When the girl's brother tries to save and defend her, he, too, is killed.
Doctor Manette has a good reputation as a young doctor, and the Evremondes virtually kidnap him from his home in order to come tend to the girl, who is suffering from some kind of lunacy, going mad from these events and routinely repeating a phrase which mean nothing to the doctor. Manette hears this story from the brother before he dies, and then he tries to save the girl
"When I returned to the bedside of the young woman, I found her raving in precisely the same order of continuity. I knew that this might last for many hours, and that it would probably end in the silence of the grave."
That is precisely how it ends. Once she dies, Manette is taken back to his home. But that is not the end of things. Manette is a good man who feels as if he must report what he knows to the authorities. It is clear that the Evremondes are watching him, for they intercept the letter he sends to the authorities and immediately swoop him up again. This time he is taken to the Bastille, where the brothers have used their power and influence to imprison him there indefinitely.
While in prison, Manette painstakingly writes this letter, as he believes the world must know what happened to him as well as to the young girl. He closes his letter with a damning denunciation:
"And them and their descendants, to the last of their race, I, Alexandre Manette, unhappy prisoner, do this last night of the year 1767, in my unbearable agony, denounce to the times when all these things shall be answered for. I denounce them to Heaven and to earth."
This matter is supremely important to Madame Defarge because the girl in the story was her sister, and she has made it her life's work to kill every seed of the Evremondes, including Charles and his family. Unknowingly, Manette has helped her try to do it.