The Catholic Church had serious issues to address during the Middle Ages, and we can see some of the concerns in Geoffrey Chaucer's The Canterbury Tales. In this piece of writing, Chaucer is quick to point out how the Church was failing the people.
Chaucer's Pardoner sells "hot" (stolen) pardons to the people, along with fake relics, such as a piece of clothing he passes off as the veil of the Virgin Mary and animal bones he insists are the bones of saints. He is not a true servant of the Church, but of himself. Chaucer points an accusatory finger also at the Friar and the Monk. He is, however, impressed with the Parson, offering praiseworthy examples as to how he lives the pious life that all the Church's servants should follow, in putting his own welfare second to that of his "flock."
Chaucer is simply writing about how those in service of the Church were often self-centered, and did very little to save the souls of the common folk; instead they cheated and took advantage of the common man. These are simply symptoms of larger issues.
With regard to the feudal system (that would disappear with the advent of longbows and dynamite), there was a struggle between "kings and popes." There was the struggle to maintain who wielded the power. Kings wanted to have power over the state and the Church, while the leaders of the Church felt that they had the power over the state and Church as representatives of God.
In the 5th Century, with the fall of the Roman Empire, the Church became the most powerful entity in the West. Struggles between political leaders (kings) and the Church began in the 10th Century.
When the plagues hit Europe, thousands of people died (and there was more than one plague). This changed the face of Europe in terms of survivors taking land. There were new struggles for leadership and control, and this impacted the Church, which preached that the plagues were a punishment against man and his sin.
Information began to arrive from other parts of Europe, which brought questions into the minds of the people, who would now be more prone to ask questions of the Church.
The absorption of Latin texts had started in the 12th century Renaissance through contact with Arabs during the Crusades, but the availability of important Greek texts accelerated with the capture of Constantinople by the Ottoman Turks, when many Byzantine scholars had to seek refuge in the West, particularly Italy.
With the advent of printing, materials were now available to the general public, no longer controlled by the Church exclusively; up until this time, the Church controlled everything that the people learned. Masses were held in Latin which preserved a sense of the mystical side of religion. This was a blow to the dominant control over the ideas of the public.
The changes brought about by these developments have caused many scholars to see it as leading to the end of the Middle Ages, and the beginning of the modern world.
It would not be too long after that Henry VIII split with the Roman Church, creating a new [Anglican] religion to give himself a divorce, and Martin Luther would nail his Ninety-Five Theses to the door of the church in Wittenburg, starting the Protestant Reformation. Inevitably, Luther would also print the Bible in German, making even more information available to the public.
This question is worthy of many books. The Middle Ages lasted a long time and many, many changes happened. I will mention a few important ones.
- There were conflicts between the Church and secular leaders about who should appoint Church leaders. This led to important changes like the creation of the College of Cardinals to elect popes. It also led eventually to the Concordat of Worms in which the Church got the right to appoint all its own bishops.
- It was in the Middle Ages that the Church started to be strict about requiring clergy to be celibate.
- There was the split between the Roman Catholic Church and what is now called the Eastern Orthodox Church.
- The Cluniac reforms made monasteries a much more important part of the Church.
In general, there were attempts to reform the Church, to make it more organized and to separate it from secular authority. However, this is a huge generalization and there is much more detail than can possibly fit into an answer of this length.