There are two distinct doctrines believed about the Bishop of Rome by Roman Catholics:
For a definition of Papal Primacy, we can turn to the ecumenical council of Florence, and also a decree from Vatican 2:
Council of Florence (1439 AD) - Session 6
6 We also define that the holy apostolic see and the Roman pontiff holds the primacy over the whole world, and the Roman pontiff is the successor of blessed Peter prince of the apostles, and that he is the true vicar of Christ, the head of the whole church, and the father and teacher of all Christians, and to him was committed in blessed Peter the full power of tending, ruling and governing the whole church, as is contained also in the acts of ecumenical councils and in the sacred canons.
Vatican 2 (1965 AD) - Lumen Gentium
25 This religious submission of mind and will must be shown in a special way to the authentic magisterium of the Roman Pontiff, even when he is not speaking ex cathedra; that is, it must be shown in such a way that his supreme magisterium is acknowledged with reverence, the judgments made by him are sincerely adhered to, according to his manifest mind and will. His mind and will in the matter may be known either from the character of the documents, from his frequent repetition of the same doctrine, or from his manner of speaking.
It's easier to work from the perspective of Papal Primacy, because it applies generally to the Pope at any time, whereas Papal Infallibility is limited by Catholics to an extremely narrow set of circumstances, such that there is only a consensus around a handful of statements that have ever qualified as Ex Cathedra in the entire history of the Church.
Therefore, we know that the Bishop of Rome is, according to the Catholic Church, the one to whom all Christians are to look to for teaching, as their Vicar of Christ.
Pope Honorius I was the Bishop of Rome from 625-638 AD. While he was Pope, he was asked for his judgment on the doctrine of Monoenergism - that Christ had one "energy" - by the Bishop of Constantinople, Sergius I (565-638 AD). In 635 AD, he responded that the doctrine that Christ had two energies, Dyoenergism, could lead to the mistaken belief that Christ had two wills. In doing so, he supported the doctrine of Monothelitism - that Christ only had one will.
This doctrine was later condemned as heretical, and the Monothelites were all anathematized in the Third Council of Constantinople - including Pope Honorius, by name:
Third Council of Constantinople (681 AD)
8 The holy council said: After we had reconsidered, according to our promise which we had made to your highness, the doctrinal letters of Sergius, at one time patriarch of this royal god-protected city to Cyrus, who was then bishop of Phasis and to Honorius some time Pope of Old Rome, as well as the letter of the latter to the same Sergius, we find that these documents are quite foreign to the apostolic dogmas, to the declarations of the holy Councils, and to all the accepted Fathers, and that they follow the false teachings of the heretics; therefore we entirely reject them, and execrate them as hurtful to the soul.
But the names of those men whose doctrines we execrate must also be thrust forth from the holy Church of God, namely, that of Sergius some time bishop of this God-preserved royal city who was the first to write on this impious doctrine; also that of Cyrus of Alexandria, of Pyrrhus, Paul, and Peter, who died bishops of this God-preserved city, and were like-minded with them; and that of Theodore sometime bishop of Pharan, all of whom the most holy and thrice blessed Agatho, Pope of Old Rome, in his suggestion to our most pious and God-preserved lord and mighty Emperor, rejected, because they were minded contrary to our orthodox faith, all of whom we define are to be subjected to anathema. And with these we define that there shall be expelled from the holy Church of God and anathematized Honorius who was some time Pope of Old Rome, because of what we found written by him to Sergius, that in all respects he followed his view and confirmed his impious doctrines.
16 To Honorius, the heretic, anathema!
This decision to anathematize Pope Honorius was affirmed at the Second Council of Nicaea, as well as the Fourth Council of Constantinople:
Second Council of Nicaea (787 AD)
We affirm that in Christ there be two wills and two operations according to the reality of each nature, as also the Sixth Synod, held at Constantinople, taught, casting out Sergius, Honorius, Cyrus, Pyrrhus, Macarius, and those who agree with them, and all those who are unwilling to be reverent.
Fourth Council of Constantinople (870 AD)
Further, we accept the sixth, holy and universal synod, which shares the same beliefs and is in harmony with the previously mentioned synods in that it wisely laid down that in the two natures of the one Christ there are, as a consequence, two principles of action and the same number of wills. So, we anathematize Theodore who was bishop of Pharan, Sergius, Pyrrhus, Paul and Peter, the unholy prelates of the church of Constantinople, and with these, Honorius of Rome, Cyrus of Alexandria as well as Macarius of Antioch and his disciple Stephen, who followed the false teachings of the unholy heresiarchs Apollinarius, Eutyches and Severus and proclaimed that the flesh of God, while being animated by a rational and intellectual soul, was without a principle of action and without a will, they themselves being impaired in their senses and truly without reason.
In addition to the councils above, he was also anathematized by name in the oath taken by every new Pope for a period of roughly 300 years:
Catholic Encyclopedia - Pope Honorius I
Pope Honorius was subsequently included in the lists of heretics anathematized by the Trullan Synod, and by the seventh and eighth ecumenical councils without special remark; also in the oath taken by every new pope from the eighth century to the eleventh in the following words: "Together with Honorius, who added fuel to their wicked assertions" (Liber diurnus, ii, 9).
So, according to the 6th, 7th, and 8th Ecumenical Councils above, the man who was Pope for a period of 13 years, was "expelled from the holy Church of God and anathematized", because he taught heresy in an official capacity, along with Sergius I of Constantinople. And, every Pope for 3 centuries reaffirmed this anathema.
How is this consistent with the doctrine of Papal Primacy? The Patriarch of Constantinople appealed to the Pope for guidance, and the Pope - "the true Vicar of Christ, the head of the whole church, and the father and teacher of all Christians" - gave him heresy, and then he was anathematized repeatedly over the coming centuries.
So, if Pope Honorius is "expelled from the holy Church of God and anathematized", who was the Vicar of Christ for the 13 years that he was Pope? Did Christ leave His flock without an infallible Vicar to lead them, for 13 years? How could the Chair of St. Peter be occupied by an anathematized heretic, for over a decade?
In summary, the fact that Honorius was anathematized for heresy shows that the Pope is not trustworthy as a source of doctrine, because if any had followed him during the time of Honorius - or in the interim of 40 years between Honorius's death and the official condemnation of his letters - they would have been heretics, who would be anathematized by later councils.
It also shows that the early Church did not believe in the modern Catholic Church's concepts of Papal Infallibility, or Papal Primacy in matters of faith and morals, which is why they showed no scruples in naming him, condemning him as a heretic, and anathematizing him, repeatedly, and as a result, leaving a 13-year gap in the Papacy.
And lastly, I thought that this quotation from James White well summed up the illogical Catholic position on the Papacy, in spite of the inconvenient facts of history, like Honorius:
James White - Failure to Document: Catholic Answers Glosses Over History
The simple fact of the matter is that no one in the days of Honorius believed in "Papal Infallibility" as it is defined today. In reality, no papal statement of the past is liable to be proven in error since it can always be argued that they "did not intend it to be a binding statement upon the entire church." So, through the wonderful use of hindsight, any errant Papal teaching can be considered non-binding, and any Papal teaching that is still in vogue can be said to be consistent with the "universal faith of the Church." It’s truly a wonderful system that actually means nothing at all, since you can never know if a current Pope’s teaching will get the stamp of approval of future generations, or will end up on the scrap-heap of "he didn’t mean that infallibly" pronouncements.