As described by Betterthan Kwora' s quote from Gibbons there was a time when the people of Rome were directly involved in Papal elections, but this ceased well before 1059. The last few popes prior to the In Nomine Domini decree had been appointed by the Holy Roman Emperor without any vote.
Thomas Greenwood (1790-1871), an English barrister and fellow of Durham University, wrote Cathedra Petrie, a multi-volume political history of the Roman Papacy.
In pages 163 to 164 of volume 10 he discusses the "indefinite and somewhat ambiguous" terms of the In Nomine Domini decree. He says that it would occur to any ordinary understanding that both the clergy, and the people, of Rome had vetoes over the Cardinal Bishops choice; and that in the event of dissent it would be for the Cardinal bishops to propose a more acceptable candidate. . However, he says, there is so little in subsequent practice to throw light on the meaning that we can infer that it was intended to deprive the populace of any role.
The decree says that the election of a pope will be ratified by the people of Rome, not that they may choose whether or not to ratify it., and makes no provision as to how this should happen, other than by accepting and recognising the decision made.
Pope Nicholas II died two years after issuing the decree, so the Papal Election of 1061 was the firs time that it was used in practice. There was no formal mechanism for ascertaining the views of the Roman people. The Cardinal bishops alone elected Pope Alexander II and crowned him at night in the San Petro in Vincoli basilica. This was because opposition in Rome made it impossible for them to get to the usual St Peter's basilica. Meanwhile prominent citizens of Rome had gone to see the young king Henry IV to request his permission to elect a pope which they did in the form of Honorius, now regarded as an antipope.
Norman soldiers, based in Italy, were brought into Rome, for a fee, to assist Alexander. On page 192 Greenwood describes a bloody engagement in the streets of Rome. In the first assault the Normans were driven back but eventually Count Richard was able to conduct Alexander to the Lateran and "with bloody hand" install him on the pontifical throne.
Whatever Nicholas meant by ratification, it is probably safe to say he was not thinking of riotous opposition.
The next papal election was in 1073. During Alexander's funeral the crowds began acclaiming Hildebrand for pope. There were only two cardinal bishops present in Rome and they agreed. In this decision, at least, the Roman people were prominent,
It would be too long to go through the whole millennium, so moving straight to the present day: the laity of Rome play no role in the election of a pope. The decision is instead taken on their behalf by representative bishops, priests and deacons from churches in the diocese of Rome and the surrounding suburban dioceses. How so? Although the cardinals who elect the pope are drawn from all over the world, every cardinal on his appointment is assigned a nominal position as titular bishop, priest or deacon of one of the churches in Rome. In this way it is, at least in theory, true to say the Bishop of Rome is elected by the clergy of Rome.
The ordinary people of Rome are required to ratify the decision only in the sense that they are obliged to recognise and accept that it has been made.