The current practice of the Roman Catholic Church is that only "cardinal-electors" are eligible to vote for the next Successor of St. Peter.
This narrows down the pool to 133 cardinals this time around, who will be given the task to listen to the Holy Spirit and prayerfully cast their ballot for the next pope.
It is a process that has centuries of tradition behind it, and it has been adjusted over the years after learning from some of the mistakes of the past.
Lay influence
One period of papal elections that we know little about is the period of the very early Church. Some traditions claim that St. Peter chose his successor, St. Linus, before he was martyred. It's possible some lay people were involved in helping pick the next successor, as it was such a small Church in the beginning -- but it is difficult to know.
We know laity were consulted in the 4th century, though not the ordinary people of the pews. The Roman Emperor wanted to be involved, ensuring that the pope would not challenge his political plans.
The Catholic Encyclopedia provides a brief summary of this period of lay influence, which lasted many centuries:
After Constantine had given peace to the Church, the Christian Roman emperors often took part in the institution of a new pope and at times their influence was very marked. From the fourth century onwards, therefore, a new force had to be reckoned with... On the recovery of their influence in the Italian peninsula, the Eastern emperors required that the choice of the electors for a new pope must be made known to the Exarch of Ravenna, who in turn forwarded it to Constantinople, and until the emperor's confirmation was received, the candidate was not to be acknowledged as Bishop of Rome. This resulted in long vacancies of the Holy See.
Eventually the Church recognized the negative results of such a process and "in 769 a decree was framed in a synod of the Lateran, that the Roman clergy were to choose as pope only a priest or deacon, and forbade the laity to take any part in the election."
What's remarkable is that despite these worldly influences on the papacy, the pope never officially contradicted Church teaching to appease an emperor or king.
Some popes have led a sinful life, or supported military campaigns of an earthly ruler, but never has one openly taught against Church teaching.
This is the promise of the Holy Spirit that the pope will speak infallibly on matters of faith and morals. The First Vatican Council dealt with this issue and proclaimed:
When the Roman Pontiff speaks ex cathedra, that is, when in exercising his office as shepherd and teacher of all Christians he defines with his supreme apostolic authority that a doctrine on faith and morals is to be held by the whole Church, through the divine assistance promised him in the person of Saint Peter, he enjoys that infallibility with which the divine Redeemer wished to endow his Church in defining a doctrine on faith and morals.
Even if a pope is chosen and influenced by a king or emperor, it is Catholic belief that the Holy Spirit protects from being taught any error on the topic of faith and morals.
