Medieval people believed in miracles. So does modern society.
How many times have you heard someone say, "It was raining so hard the morning of my party, but it cleared up at noon, it was like a miracle," or "Every new baby is a miracle come into the world," or "His car was totaled, but he was fine, it was a miracle," or "At dawn the world is all still and bright, like a miracle."
A miracle literally means something marvelous and wonderful, which is how modern society usually uses the term. Medieval people used it this way too. For example, they would speak of a noble lord who had been stealing a monastery's flocks as "miraculously" repenting, or a church architect "miraculously" finding timbers just the right size for the new beams.
The word's more specific meaning, then and now, is something that cannot be explained naturally. The modern Catholic church requires verified miracles to declare someone a saint. That generally means a healing where the doctors have said there is no hope of a cure. This criterion for sainthood came in during the twelfth century. A person who had lived a saintly life would no longer be declared a saint (as s/he would have been for the first millennium of Christianity) without some well-documented post-mortem miracles.
As I discussed earlier, it is important not to label medieval belief in miracles "superstition." People then certainly did not believe every purported miracle. Some eventual saints, like Bernard of Clairvaux, took a while to be officially recognized because his supporters didn't have enough well-documented healings.
Although the assumption now is that "over-educated elitist university types" are least likely to have strong religious beliefs, which are found instead in the simple, hard-working folk of the heartland, this assumption was upside down in the Middle Ages.
Instead, it was the best-educated in the Middle Ages who clung the most strongly to religion and who battled most desperately any doubts that might come wandering in. This is because the smartest minds of the age all went into theology. Their religion and theology were certainly not simple, and nobody believed the Bible was meant to be read literally, but there are plenty of indications that the most devout churchmen and churchwomen intermittently found themselves wondering if God cared a farthing for His creation.
This carving, from the twelfth-century church of Saulieu, shows the story of Balaam from the Old Testament. He was understandably startled when his donkey stopped dead and started to speak. This is because the donkey, but not Balaam, could see the angel with a sword ahead of them. Balaam was a doubter, the kind of doubter the monks wanted to remind themselves not to be.
The biggest doubters were among the peasants. Such doubters appear in all the accounts mocking the saints, making fun of relics, saying, "How can some old bone heal, Mr. Monk? I could get better results with a stone I picked up in the field!" Then they'd laugh and try to steal the offering.
Because Christianity was the dominant belief system they'd essentially adhere to it, get their children baptized, try to get Grandpa last rites, but it was quite startling when they were told that they were supposed to go to church and partake in the mass at least once a year, whether they needed it or not.