We take calendars for granted. Doubtless there is one hanging on a wall somewhere near you. We write appointments on the calendar--"Billy to the doctor at 4:15 on the 21st."
Medieval people had the same basic calendar we do (365 days spread over 12 months, the same ones with 30 or 31 days, February always gets shorted but does get an extra day every 4 years). This is and was your basic Julian calendar. But, just as medieval people thought about telling time differently (as I have discussed earlier), so they thought of calendars differently.
They would not have a page for each month with an attractive picture (flowers, a rural scene, puppies) next to a grid of little boxes, each with a day number. (This is an attractive rural scene, in case you couldn't tell.)
Instead each day of the month would be listed underneath the others. Next to it would be the saint whose special day it was, whose anniversary would be celebrated that day. Although modern Catholicism has a saint for every day of the year, a lot of medieval calendars wouldn't have anyone special to commemorate on many days. Unusual events might also be noted, such as an eclipse or a flood.
One of the most important things to note on a calendar was Easter. In the Middle Ages, Easter was by far the most important religious holiday, unlike today when it has to take a back seat to Christmas. Easter is now (as then) the first Sunday after the first full moon after the Equinox. As you can imagine, it moves around a lot. Exactly when the Spring Equinox falls is open to discussion, and do you count the first full moon as the day whose evening sees it rise or the following day? In 2016, Greek Orthodox Easter fell in May, a month after Latin Easter. Medieval people had all sorts of charts and tables trying to show when Easter would be next year.
We assume the new year starts in January. Some medieval people did as well. But depending on the region, the new year might start at Christmas, at Easter, or even at the Incarnation (March 25, nine months before Christmas, when Mary became pregnant). Easter was the most common, which meant that if Easter was early one year and late the next, a "year" might have almost two whole Aprils in it, one at each end. Sometimes, especially in Italy where the cities were in fierce competition, one could be off by one or two years just by walking a short distance from one city to another.
We count days of the month by the day, 1st, 2nd, etc. Medieval people, like the Romans, were more likely to refer to the kalends or the ides. The kalends was the 1st (it's where the word "calendar" comes from). So a medieval document might be dated by "the second kalends of June," meaning the next to last day of May. Ides fell in the middle of the month (Caesar was famously assassinated on the Ides of March, or March 15).