Short answers, pulled from the story.
A calendar is a system of organizing days by giving names to periods of time, typically days, weeks, months, and years. The word can also mean the physical paper record of such a system, or a list of planned events such as a court calendar.
The word calendar comes from the Latin kalendae, the name for the first day of the month in the Roman calendar, related to the verb calare, meaning to call out, a reference to the calling of the new moon. The Latin calendarium meant account book or register, because debts were collected on the calends of each month.
The Gregorian calendar was introduced in 1582 as a refinement to the Julian calendar, amounting to a 0.002 percent correction in the length of the year. The last European country to adopt it was Greece, in 1923.
Solar calendars are based on the apparent motion of the Sun and the seasons, while lunar calendars number days within each lunar phase cycle and drift against the seasons. Lunisolar calendars combine both, adding an extra intercalary month as needed to realign the months with the seasons, as the Hebrew calendar does with a 19-year cycle.
Julius Caesar reformed the Roman calendar in 46 BC, creating the Julian calendar. It no longer depended on observing the new moon and instead followed an algorithm of introducing a leap day every four years, which separated the calendar month from the lunation.
The Islamic or Hijri calendar is a lunar calendar of 12 lunar months in a year of 354 or 355 days. Its epoch is the Hijra, corresponding to AD 622, and its annual drift of 11 or 12 days means the seasonal relation repeats approximately every 33 Islamic years.
The Egyptians appear to have developed the first solar calendar, using the annual sunrise reappearance of the Dog Star, Sirius or Sothis, which coincided with the flooding of the Nile River. Their calendar had 365 days divided into 12 months of 30 days each, with 5 extra days at the end of the year.