It's the "Mamertine" prison, on two levels below the Chapel of the Crucifix that attract the attention of most visitors, and luckily those levels have a separate entrance which is open almost daily. (9am-12:30 and 2:30-6pm, April through September, and 9am-noon and 2-5pm, October through March -- admission is free, but a small donation is expected by the monk/doorkeeper.) The two levels were already in ancient times called the "carcer" and the "tullianum".
And according to legends already circulating in republican Rome, the carcer was built as a prison by Rome's fourth king, Ancus Marcius and then the lower chamber, the "tullianum", was dug below the carcer during the reign of the sixth king, Servius Tullius, and took its name from him. Archeological and physical evidence clearly contradicts the legends. The tullianum chamber clearly existed before the upper carcer. It originally was circular and probably was constructed as a cistern around the still flowing spring in its floor (the Latin word tullus means spring. When the upper chamber was added to convert the building into a prison, the lower structure was truncated, and entry to it was blocked except for a hole in the floor of the upper chamber, through which prisoners were lowered. The time of the construction of both chambers is questionable, but it is clear from literary references that both levels already existed by the middle of the Republican period. It is probable that the floor of the tullianum was at or slightly below ground level when it was built around the 4th century BC and that, after the ground level had risen somewhat, the carcer was built above the truncated tullianum, also at or near its own contemporary ground level in the 2nd century BC. The carcer is a dozen of so several steps below today's street level.
The attraction to this place for most visitors is that this is the prison where St. Peter was supposedly kept while awaiting his execution in Rome, and thus would have been the scene of one of his angelic visitations. According to the legend, Peter was freed from his chains by the angel and was fleeing south down the Appian Way when he met a man he recognized as Jesus, who was heading into Rome. After a short conversation, Peter decided to accompany the man back to Rome to accept martyrdom. The church of "Dominus, Quo Vadis?" marks the spot on the Appian where Peter asked the man, "Lord, where are you going?" The chains, centuries later, were miraculously conjoined with the other set of chains that bound Peter when he was imprisoned in Caeasarea Maritima in Palestine, and they are in the church of St. peter in Chains on the Esquiline Hill. So goes the legend, despite the fact that many scholars accept the possibility that Peter never visited Rome.
Whether or not Peter was ever an inmate of the Mamertine prison was the subject of heated theological and ideological debates, mostly between Protestants and Catholics, up to the early years of the 20th century, and research sparked by those debates revealed several interesting facts. Among them, it became clear that no ancient Roman would have recognized the name "Mamertine" -- that was a Medieval name attached to the site, probably to tie the prison to Roman legends surrounding Peter. The ancient Romans, in their usual prosaic style, simply called the site "carcer", which we usually translate as "prison". The Latin word, apparently derived from Etruscan, originally meant an enclosed space or pen, but it is clear that later Latin usage almost always referred to this particular structure in the forum, just north of the Sacra Via and near the Senate's meeting hall, the Curia.
The "carcer" was, in fact, the only prison of ancient Rome, and it was never designed as a place of long-term imprisonment for the punishment of common criminals -- immediate corporal punishment, exile, or the arena was their fate. Betrayal of the state (for example, opening the gates to surrounding enemies) might get you thrown off the Tarpeian Rock. Being an enemy of the Emperor, or just owning a property he coveted, might get your name on a list of proscribed ("written down") persons that anyone could kill on sight. Imprisonment in the carcer was reserved for important state prisoners -- often foreign leaders who had been unwilling exhibits in triumphal parades -- while they waited the few hours for their almost inevitable executions. For them, the carcer was a short-term "death row", a holding area, and not a place of long term imprisonment. The actual killing would be carried out precisely at the moment when the victorious Roman general would dedicate captured treasures in the temple of Jupiter Optimius Maximus on the Capitoline. The death of the prisoner was also dedicated to the god in a long-lasting holdover of rites of human sacrifice. Jugurtha, a North African successor of Hanibal was executed in the Carcer as was Vercingetorix, the leader of the Gauls. Other state prisoners were executed either inside the carcer or, if a public spectacle was deemed advisable, in front of the Senate building.
Cicero also had some of the Catalinian conspirators "incarcerated" (that is, put in the carcer) before unconstitutionally and unceremoniously having them executed there. Cicero himself was executed years later after making himself an enemy of Octavian (later Augustus). Technically, he died after being proscribed by the Second Triumvirate: Octavian, Marc Antony, and Lepidus. But he did not rate immurement in the carcer. Instead, he was simply hunted, captured, and killed at Caieta (modern Gaeta), on the coast between Rome and Naples. His head and his famous expressive gesticulating hands were displayed on the rostrum in the Roman forum.
The best Internet web side on the Carcer/Tullianum and surrounding areas is provided by the Suprintendenza Archeologica di Roma (only in Italian) at http://www.archeorm.arti.beniculturali.it/sar2000/nuovi_scavi/default.htm
Lacus Curtius has the Platner and Ashby "Rome Topographic Dictionary citation", in English, at http://www.ukans.edu/history/index/europe/ancient_rome/E/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/.Texts/PLATOP*/Carcer.html and the Christian Hulsen citation, from his "Il Foro Romano - Storia e Monumenti", in Italian, at http://www.ukans.edu/history/index/europe/ancient_rome/I/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/Forum_Romanum/.Texts/Huelsen*/2/20.html but both are somewhat out of date. (The entry in Richardson's "New Topographical Dictionary of Ancient Rome", published in 1992 and not available on the Internet, is not much better.)
Basic information on the St. Joseph church is at http://home.online.no/~cnyborg/giuseppefalegnami.html and some old prints and modern pix are at http://members.tripod.com/romeartlover/Vasi42.html.
More pix, some of the interiors of the carcer and tullianum, are at http://wings.buffalo.edu/AandL/Maecenas/rome/mamertine/thumbnails_contents.html
To see what the carcer looked like in ancient times, visit http://www.unicaen.fr/rome/geographique/carcer.html. It's the low walled structure in the foreground, not the adjacent temple.
Cicero's life and death is outlined
succinctly at http://www.crystalinks.com/cicero.html