To understand the council, one must get out of one's head any nonsense about the Greek East as some kind of opposition to the Latin West. In the seventh century, there were no such sharp lines. In Italy, Greek and Latin rites were all jumbled together -- Italy in a north-ish and west-ish was largely Latin due to the influence of Rome and Milan, but the church in southern Italy was very decentralized, and churches were generally founded by local patronage, and while each church would generally be either Byzantine or Latin rite, it was just a matter of whichever the local patrons happened to decide that they wanted. There was a slow latinization under the southern Lombards, but it was very slow and barely noticeable at this early a date. And Italy was not entirely under Lombard rule; the other major power was the Exarchate of Ravenna, which was firmly Byzantine. Nor were the boundaries between the two powers neat and clean, as you can see from this map of the Lombard/Byzantine division of Italy at the end of the sixth century, shortly after the Lombard invasions:
Nor, for that matter, did the boundaries necessarily mean much; being in Lombard territory still could mean considerable direct Byzantine influence, depending on the particular political situation. Moreover, from 642 to 752 (from John IV to Zachary) there were eleven Popes whose native language was Greek -- either from Sicily, or from Syria, or from Greek-speaking parts of Italy, or (in one case) from Ephesus. The only exceptions, I believe, were the Popes who had been born in Rome itself. What is more, Muslim invasions both east of Italy (Syria) and west of Italy (Sicily) had led to a very significant influx of Greek-speaking refugees. There was a broader Latin world, to the north and to the west, over which Rome had extraordinary influence; but Rome itself in the seventh century lived in a world that was in a great many ways more Greek than Latin, more Eastern than Western.
The Emperor, Heraclius, was faced with a serious set of problems; he was very concerned with recovering territory that had been lost to the Persian Empire and preventing yet more territory from being lost, and the disunity between Christians who followed Chalcedon and those who rejected it was interfering with the formation of a unified front against the Persians. With the support of Patriarch Sergius of Constantinople, he attempted to push a compromise position: if the Monophysites would accept the formula that Christ had two natures, divine and human, the Chalcedonians would accept the formula that Christ had one operation or will (Monothelitism). It seemed to many to be a viable solution to a serious problem, a way to bring peace to a fight of which everyone was tired. Alexandria and Antioch signed on. The East was coming together -- or so it seemed. Nobody had reckoned on St. Sophronius. Sophronius was a widely respected monk; in 633 he traveled to Alexandria and to Constantinople in an attempt to convince their respective patriarchs that the compromise was unacceptable. He failed. But the next year he was elected Patriarch of Jerusalem, and from that position he began actively opposing the Monothelite compromise.
So now began the maneuvering. Patriarch Sergius wrote to the Pope at the time, Honorius I, insisting that the relevant phrase, one operation (energy) of Christ, had been used in a letter by Patriarch Mennas to Pope Vigilius (the letter was actually a forgery, but Sergius was not in a position to know this), and that the compromise was making it so that people who had before repudiated the words of Pope St. Leo I to Chalcedon were now singing his praises: unity was being achieved. However, he was willing to drop the precise formula he had been using, if Sophronius would also drop his. In the face of this argument (it is explicitly the unification of the East that the Pope found impressive), Honorius replied that this was probably for the best, and affirmed Sergius's position, or at least something like it; later, people will suggest that Honorius took Sergius to be denying that Christ had two contrary wills, and possibly that's what he was thinking, although it's hard to say. But it still made the dispute one of Constantinople, Alexandria, Antioch, and Rome against Jerusalem. Or so it seemed. A synod met in Cyprus with representatives from all sides; and it came down in Sophronius's favor. And with that the whole compromise began to fall apart.
But Sergius was not deterred. The formula could be tweaked; and he tweaked it. This modified version is what would become known as the Ecthesis, and Sergius got Heraclius to sign off on it -- Heraclius would later say reluctantly, but how much this is actually true is difficult to say. All this happened in 638. Why 638? Because St. Sophronius died in March of that year. Patriarch Sergius would die in December of the same year -- but by the time he had done so, all four eastern sees had signed on to the new compromise, in part because Sergius was now able to use Honorius's letter. Eastern unity was now within reach. Or so it seemed.
Honorius had died in October 638, after what had seemed at the time to be a quite competent and relatively uneventful thirteen years of papal rule. Three days after his death, Severinus was elected Pope. But at the time, it was expected that no pope would take office until he received confirmation documents from the Emperor. Heraclius made it a requirement that Severinus could not receive the documentation until he signed on to the Ecthesis. Severinus had already refused to do so, but after some negotiation they were able to get Heraclius to agree to confirm the election on condition that they would show the Ecthesis to Severinus and ask him to sign it if he agreed with it. So the Exarch of Ravenna was to make sure that Severinus did sign it and then issue the confirmation in the Emperor's name. Severinus, however, refused. More negotiations followed and finally in 640, Heraclius, who was very ill, gave in, and confirmed the election. Severinus immediately called a synod and condemned the Ecthesis. Because of the long negotiation to get Imperial confirmation, which had led to his practically being a prisoner in the Lateran Palace for a year, it was about all he was able to do, as he died two months after the confirmation. But the momentum was now set: John IV (from Greek-speaking Dalmatia), Severinus's successor, also refused to accept the Ecthesis. And after John IV came St. Theodore I, a Greek-speaking Syrian from Jerusalem, and with him the foundations of the Third Council of Constantinople began to be laid. Theodore was very much against Monothelitism, and attempted to depose Patriarch Paul II of Constantinople for his support for it.
Of St. Maximos the Confessor in his early life we know relatively little for certain, although there is excellent reason to think that he was from Constantinople, served for a while as a civil servant there, and, after he became a monk, studied under Sophronius. He became involved in the Monothelite controversy while in Carthage, and he began a correspondence with Pope Theodore in 646, shortly before journeying to Rome to meet the Pope in person. He and Theodore planned a council to handle the matter. But it was not just any ordinary synod. Maximos's plan was for the Pope to call an ecumenical council. This was a bold move, because all ecumenical councils up to that point had been called by Emperors. Maximos was the one who did much of the practical work of planning for the council. Two things, however, would complicate matters.
The first was that in 648, the Emperor Constans I issued a decree called the Typos, which prohibited any discussion of Monothelitism at all -- quite literally, it commanded by law that everyone should go back to the way things were before the dispute arose. The second was that Theodore died suddenly in 649, before the council could actually be convened. He was succeeded by St. Martin I, who had spent much of his career in Constantinople. This put the council temporarily under a bit of doubt, but, as it happened, Pope Martin was of the same mind as Pope Theodore had been, and he did not wait for any imperial confirmation of the election; he called the council into session almost immediately, and the council began almost immediately because almost all of the planning had been done for it. Thus came about the Lateran Council of 649, the almost-ecumenical council. While there was a judicious selection of Latin bishops from Italy and North Africa, almost all of the major movers of the council were Greek-speaking, and perhaps as many as a quarter of all the bishops, as well. The council claimed that the bishop of Rome had full authority to eliminate heresy, and it condemned both the Ecthesis and the Typos. Martin promulgated the acts of the council by an encyclical claiming that the council had ecumenical authority.
Needless to say, Emperor Constans empowered the Exarch of Ravenna to arrest both Pope Martin and Maximos. Both were eventually caught in 653. Martin was exiled without trial and died in 655. Maximos was tried for heresy, found guilty, and as punishment had his tongue and right hand cut off; he was then exiled, and died in 662. The quieter St. Eugene I did not push the matter of the council with Constantinople, nor did St. Vitalian afterward, although both were opposed to Monothelitism. Adeodatus II did not involve himself with the question at all, and we have very little information about what Domnus, the next pope, did on this matter. But in 678, St. Agatho, from Sicily, became pope, and he was of a more energetic mettle.
And the timing was right, as well. The Emperor Constantine IV had a number of successes against the Muslim armies that had arisen and distracted imperial attention from the question of schism, and thus having a space to breathe, he wrote a letter to Pope Domnus proposing an ecumenical council to resolve the matter. Domnus was already dead, but Agatho wasted no time at all; he had synods throughout the West called to discuss the matter and arranged to send a large delegation -- the largest delegation of Western bishops that had to that point ever been sent to an ecumenical council -- and the Third Council of Constantinople officially opened in 680. A letter from Agatho was read at the council that came down strongly against the Monothelite heresy; it was accepted by the council and helped to consolidate the opinion of the council against Monothelitism. The council condemned Monothelitism and the major Monothelites, including Pope Honorius. Sophronius is explicitly upheld as orthodox in the Acts of the Council. The council held under Pope Martin is explicitly referred to in the letter of Agatho, which was accepted as orthodox by the council.
Pope Agatho did not live to see the triumph; by the time the decrees of the council had been sent back to Rome, he had died, after less than three years as Pope. He was succeeded by St. Leo II, who was Pope for less than a year, but who confirmed the decrees of the Third Council of Constantinople and arranged to have them promulgated throughout the West. Leo was very careful to explain in his letters to bishops that he took Honorius to be condemned not for heresy as such but for negligence in opposing it. Leo's successor, St. Benedict II, would do further work in making sure that the West was all on the same page with regard to the council. In the East, the council had fewer problems than ecumenical councils had usually had before, and became widely accepted very quickly.
The Third Council of Constantinople had as its primary and immediate effect the consolidation of Chalcedonian orthodoxy. But the council often shows up in other contexts too -- most notably in discussions of papal authority. There are two reasons for this. The first is the condemnation of Pope Honorius, by a council that Catholics accept as orthodox. The condemnation is quite harsh, as well:
And in addition to these we decide that Honorius also, who was Pope of Elder Rome, be with them cast out of the Holy Church of God, and be anathematized with them, because we have found by his letter to Sergius that he followed his opinion in all things and confirmed his wicked dogmas.
The reason for the condemnation seems to have been that Macarius of Antioch, one of the Monothelite holdouts, repeatedly appealed to Honorius in his defense.
On the other hand, the council accepted as orthodox the letter of Agatho, which makes very strong claims about the Roman see, e.g.:
Therefore the Holy Church of God, the mother of your most Christian power, should be delivered and liberated with all your might (through the help of God) from the errors of such teachers, and the evangelical and apostolic uprightness of the orthodox faith, which has been established upon the firm rock of this Church of blessed Peter, the Prince of the Apostles, which by his grace and guardianship remains free from all error, [that faith I say] the whole number of rulers and priests, of the clergy and of the people, unanimously should confess and preach with us as the true declaration of the Apostolic tradition, in order to please God and to save their own souls.
It's doubtful that the Greeks paid much attention to such claims, but Rome has always taken things like this seriously, as more than mere rhetoric. Prior to Agatho, the popes had already considered the case of Honorius; John IV, for instance, had argued, when opposing Monothelitism, that Honorius was simply confused, and thought that Sergius was condemning the notion that Christ had two opposed wills. When Leo II affirmed and promulgated the decrees of the III Constantinople, he also made very clear that he took it to be condemning him not for teaching heresy but for failing to oppose it and thus giving it encouragement. One should not get the wrong idea about this -- Leo's condemnation of Honorius for failing to do his duty in preserving the faith is quite as sharp as anything put forward by the council, and the papal oath that it was common for new popes to take included at some point a condemnation of Honorius for assisting heretics. But it's also clear that they at no point saw this as inconsistent with continuing to affirm, as Agatho puts it, that the Roman see remained "free from all error". The case became an issue, however, in the Reformation, since Protestants regularly appealed to Honorius as an example of papal heresy; the Counter-Reformation was not able to come up with a unified response to this argument, but the conclusion that has since come to dominate is that Honorius was not in fact teaching ex cathedra, but simply stating a private opinion, albeit in official correspondence.
Regardless, the Third Council of Constantinople stands as the great achievement of the Byzantine West, and one that could arguably have never come about except by the impetus of people who in culture were both Western and Greek. The Latin West knew almost nothing about it. The Greek East could do almost nothing about it. But the Greek West both understood the problem and laid the foundations for solving it.