The answer to this question can be found in Book II Chapter II, when Jerry is given a message to take to Mr. Lorry at the Old Bailey, which is a court where criminals were judged and sentenced. The narrator of this tales goes on to explain what this location is so famous for:
For the rest, the Old Bailey was famous as a kind of deadly inn-yard, from which pale travellers set out continually, in carts and coaches, on a violent passage into the other world: traversing some two miles and a half of public street and road, and shaming few good citizens, if any.
Note how Dickens compares the Old Bailey to an inn, except this inn has only "pale travellers," who are setting off on a journey to "the other world" with a "violent passage." Criminals were tried at the Old Bailey and sentenced there, and those that were deemed to have committed a crime serious enough, were sentenced to death. When they were transported from the Old Bailey to the place of their execution, they were transported through the streets so that the residents of London could see them. This description strikes an ominous note in the trial even before the reader finds out who is actually being tried, and it foreshadows the second trial in the book which has much more tragic consequences than the first.
To this day the Old Bailey (located on Bailey Street in London) is the site of two famous courts. Court One has been the scene of the trials of many infamous criminals; Court Two is a high-security court where terrorist trials and such take place. In A Tale of Two Cities, a novel set in the 1780s, the Old Bailey is a fearful place because it is a famous "kind of deadly inn-yard" from which many depart for the grave. It is known for its pillory, and it has a whipping-post. There, too, are many "transactions in blood-money." Punishments are dealt, and in this place, "Whatever is is right."
After one of the ancient clerks at Tellson's bank sends Jerry Cruncher on an errand to the Old Bailey, Jerry becomes extremely nervous. When he arrives, Jerry asks in a whisper, "What's coming on?" He is told that there is a treason case in which the guilty man will be drawn and quartered. This man is Charles Darnay, who is clearly frightened by the hungry crowd.
Jerry Cruncher licks the rust from his fingers and tries to follow the workings of the trial of the Frenchman Darnay, who is accused of divulging secrets to the king of France (Louis XVI). Further, Cruncher notices a highly agitated young lady and her white-haired father who, he is told, are witnesses against him, yet Jerry has noticed that the young lady displays great pity for the prisoner. This information draws Jerry's attention as he waits for the verdict.
It is interesting that Charles Dickens introduces early in his narrative the images of prisons. Having drawn parallels between England and France in his initial chapter, the chapter on the Old Bailey prepares the readers for the French counterpart of the Conciergerie, where the aristocrats await death by guillotine.