What authority did the Catholic Church have over European monarchs from 900–1450?

The height of Papal Power was the penitence at Canossa by the then Holy Roman Emperor, Henry IV who waited three days in the cold outside the castle of Canossa (where the Pope GregorY VII was staying) for absolution. This occurred because the Catholic Church was much like the Holy Roman Empire, that is "neither Holy, nor Roman, nor an Empire). For all that, three of the seven electors of the Holy Roman Empire were archbishops, which is to say that the church had great sway with the greatest "Empire" of Europe. In this regard, the Catholic Church was a "superpower."

LangLang C had a point when she pointed out that the church was also what we might call nowadays a "paper tiger." That is, it had power over an emperor (the "Holy Roman" one), but less over mere "kings," whose powers were less dependent on church authority. Here, as you point out, the church had more of a mixed record. One Pope got King John of England to relent about the Archbishop of Canterbury, and King Alfonso of Portugal to "promise" to change his ways, and Frederick II, Holy Roman Emperor "relented" after three excommunications, while other Popes were less successful with other monarchs.

Still, the Popes appeared to have power that was "super" for their time (early second millennium). This was before the age of absolute monarchs, which is to say that the Popes did not have the power of the 17th and 18th century kings.

See this as an addendum/answer to @LangLangC (I ran out of characters, otherwise i'd post it as a comment).

Well I don't think it's that clear cut. First of all, the papal state itself was quite the power in Italy, which at the time was a very important region. Italy in the age of communes was among the wealthiest regions and one of the first to recover economically from the fall of Rome. Within the Papal state, the Pope wielded absolute secular power and by that measure, was even one of the first absolute monarchies in the modern sense. One such place was also the kingdom of Jerusalem, albeit short lived and not as centralized.

Secondly, such affairs as the walk to Canossa have very strong political power and implications, so much so that it was a setback for the efforts of centralization that the HRE never managed to fully recover from until its dissolution by Napoleon. The Catholic church could directly and indirectly serve as a stabilizing or destabilizing factor within a country. It could support rival centers of power and bolster legitimacy of its favorite factions.

One could regard the churches and confessionals that Catholicism had as a sort of early intelligence agency, albeit a very independent one. It often gave the church more insight into personal and political situations than its distance would normally allow.

Even after some emperors got away with investiture, the papacy still held investiture rights in other parts of Europe (NB: bishops had secular power at the time too!). So it was able to directly place prince-bishops and other secular or semi-secular posts well outside its territory. Simony, on the other hand, could provide funds and influence with local nobility in exchange for such bishoprics. Monasteries and Monastic orders were (and still are!) among the greatest property Moghuls in Europe, often having more land and income at their disposal than powerful Counts and Dukes.

The church had judicial power. Much like Sharia law today, the Catholic church held sway over many law-making bodies in Europe. This went as far as the Inquisition being able to hold independent trials and order local executive power to carry out the sentence.

The church had geopolitical soft power. Often the church was consulted to broker peace and trade treaties, seek legitimacy for some cause or the other, or to arrange marriages and other forms of alliance between countries. Think of how Inter Caetera sparked the treaty of Tordesillas.

The Catholic Church was a powerful organisation. But it wasn't a secular superpower. That is quite evident when we look at the precarious situation pope Leo found himself in, when previous to the coronation he had to flee his home turf in Rome to Charlemagne. Leo just survived an assassination attempt and made a deal with Charles, the secular power of the time: "I elevate your status and provide additional legitimacy, you save my very skin?"

Then we seem to have a misunderstanding regarding the word, process and consequences of "excommunication".

While most popes of that time were trying to powermonger ever more authority for them, they mostly failed. Precisely because excommunication was just a disciplinary measure within the church. For a Holy Roman Emperor, that was a big problem. Without papal support they were just German kings. But excommunication is only indirectly an attempt to undermine the authority and legitimacy of secular princes.

Popes also couldn't just act on whim and trump anyone they didn't like. For a political struggle a poena ferendae sententiae was developing quickly into a tightly regulated part of canon law. Popes that tried to bend the law according to their will usually found out that despite claims of spiritual superiority their actual reach might be much shorter.

Much exemplified are these relationships in a pyrrhic papal victory at Canossa. The pope Gregory used his excommunicational powers and made the emperor beg for forgiveness. But it ended with the pope being deposed by Henry, and church and state going much more seperated ways already after 1077.

The list of powerful people excommunicated is also less informative in that regard. The process itself is church internal. Much more interesting is when the popes tried to directly interfere, like in actually declaring a ruler as "deposed".

The papal deposing power was the most powerful tool of the political authority claimed by and on behalf of the Roman Pontiff, in medieval and early modern thought, amounting to the assertion of the Pope's power to declare a Christian monarch heretical and powerless to rule.

That concept sounds impressive.

But if we look at the Wikipedia list we see that it starts with Gregory vs Henry:

'I now declare in the name of omnipotent God, the Father, Son, and Holy Spirit, that Henry, son of the emperor Henry, is deprived of his kingdom of Germany and Italy'

the play went on to

His deposition still in effect, Henry was forced into civil war with Duke Rudolph of Swabia. Gregory levied a second excommunication against Henry, who ultimately won the civil war, invaded Rome, and forced Gregory to flee, replacing him with Antipope Clement III.

Then we might look at how the even very formalities involved limited this power when in 1228 another pope Gregory tried this spiel on Frederick II. Excommunicated him twice, the second time for going onto a crusade while excommunicated, technically 'incapable' of doing so, doing it anyway. Pope and church circles got very angry, Frederick got crowned King of Jerusalem.

When the old pope calls a council to make such a decision official, opponents like Pisa captured the envoys necessary.

Later

A new pope, Innocent IV, was elected on 25 June 1243. He was a member of a noble Imperial family and had some relatives in Frederick's camp, so the Emperor was initially happy with his election. Innocent, however, was to become his fiercest enemy. Negotiations began in the summer of 1243, but the situation changed as Viterbo rebelled, instigated by the intriguing local cardinal Ranieri Capocci. Frederick could not afford to lose his main stronghold near Rome, so he besieged the city.

Innocent convinced the rebels to sign a peace but, after Frederick withdrew his garrison, Ranieri had them slaughtered on 13 November. Frederick was enraged. The new Pope was a master diplomat, and Frederick signed a peace treaty, which was soon broken. Innocent showed his true Guelph face, and, together with most of the Cardinals, fled via Genoese galleys to Liguria, arriving on 7 July. His aim was to reach Lyon, where a new council was being held since 24 June 1245.

Despite initially appearing that the council could end with a compromise, the intervention of Ranieri, who had a series of insulting pamphlets published against Frederick (in which, among other things, he defined the emperor as a heretic and an Antichrist), led the prelates towards a less accommodating solution. One month later, Innocent IV declared Frederick to be deposed as emperor, characterising him as a "friend of Babylon's sultan," "of Saracen customs," "provided with a harem guarded by eunuchs," like the schismatic emperor of Byzantium, and in sum a "heretic."

The Pope backed Heinrich Raspe, landgrave of Thuringia, as rival for the imperial crown and set in motion a plot to kill Frederick and Enzo, with the support of the pope's brother-in-law Orlando de Rossi, another friend of Frederick. The plotters were unmasked by the count of Caserta, however, and the city of Altavilla, where they had found shelter, was razed. The guilty were blinded, mutilated, and burnt alive or hanged. An attempt to invade the Kingdom of Sicily, under the command of Ranieri, was halted at Spello by Marino of Eboli, Imperial vicar of Spoleto.

Innocent also sent a flow of money to Germany to cut off Frederick's power at its source. The archbishops of Cologne and Mainz also declared Frederick deposed, and in May 1246 Heinrich Raspe was chosen as the new king. On 5 August 1246 Heinrich, thanks to the Pope's money, managed to defeat an army of Conrad, son of Frederick, near Frankfurt. Frederick strengthened his position in Southern Germany, however, acquiring the Duchy of Austria, whose duke had died without heirs. A year later Heinrich died, and the new anti-king was William II of Holland.

Between February and March 1247 Frederick settled the situation in Italy by means of the diet of Terni, naming his relatives or friends as vicars of the various lands. He married his son Manfred to the daughter of Amedeo di Savoia and secured the submission of the marquis of Monferrato. On his part, Innocent asked protection from the King of France, Louis IX, but the king was a friend of the Emperor and believed in his desire for peace. A papal army under the command of Ottaviano degli Ubaldini never reached Lombardy, and the Emperor, accompanied by a massive army, held the next diet in Turin.

It is evident that "it's the economy" all the way here already. Those having more money have more soldiers. While the Catholic church was very rich, it spend quite a bit on non-military buildings, art etc.

This culminated in the early 1300s. Philip of France wanted to tax the church, pope Boniface world domination. In this situation the king was excommunicated, but the pope got 'a slap in the face', lost the struggle, died a few weeks later, and the entire papacy was moved to Avignon.

So, theoretically the pope was extremely important. Practically, secular power, with 'boots on the ground', mostly won out. It was certainly better to have the pope on your side. But if any pope started to have ideas incompatible with those of kings, church law was a often a feather, and thus a much weaker weapon than those made from actual steel.

The coronation of Charlemagne was important, but Charlemagne had the upper hand in his relationship with Leo III. Eastern Roman emperors were crowned by the Patriarch of Constantinople. No one imagined the Patriarch was superior to the emperor. Nor should you imagine that the powerful Charlemagne considered himself anything but superior to the (quite literally) mangled pope that he had rescued from a mob. The coronation of Charlemagne was a major step in the East/West divide (which had begun half a century earlier when Gregory II, Gregory III, and Zachary had split with the East over iconoclasm).

Even among the kings you list, Philip the Fair had the last laugh over Boniface VIII, having him taken prisoner, and subsequently controlling the papacy itself to a great extent, with Clement V being resident in France and doing Philip's bidding in suppressing the Knights Templar. This led a century later to the Great Western Schism and to Conciliarism, which were perhaps the lowest ebbs of papal authority in history, and correspond to the last seven decades of the period you cite (1378-1449)--not that the previous seven decades under the thumb of the French at Avignon had been any great shakes for the papacy either...pretty much erasing the last 150 of your 650 years. Even earlier, there was a strong tradition of Christian-positive clerical-negative thought in Europe, including suggestions, if I recall correctly, by authors like Wolfram (early 1200s) that laymen hear each others' confessions because priests couldn't be expected to understand the actions of military men.

Some popes, like Innocent III, really were strong players in international affairs. Others were manipulated by more powerful kings. The church was indeed very powerful, but more in the sense that the United Nations is powerful today. The Pope's opinion carried a lot of weight, like a Security Council resolution does, but exactly how much weight depended on the disposition of the audience and their military strength. It also had quite a bit to do with the political astuteness of the pope himself.

The popes even had trouble with the religious orders; see for example John XXII's famous quarrel with the Franciscans over Apostolic Poverty. And there was no shortage of princes who provided shelter to papal antagonists like John XXII's nemesis William of Ockham. Martin Luther wasn't close to being the first papal opponent who found secular help.

I should add that you might perhaps also take a look at the long string of antipopes who (along with the popes opposed to them) were the playthings of medieval kings and emperors. The list of anti-kings who were the playthings of medieval popes is rather a lot shorter....