A calendar can be any system of measuring long periods of time, including days, months and years. In Human culture, the term could also refer to a series of visual chart, usually representing months, used to keep track of such calendar.
In the ancient past, a being known as Kukulkan gave the Mayan civilization on Earth a remarkably accurate calender, according to which the Mayans were supposed to build a city awaiting his return. (TAS: "How Sharper Than a Serpent's Tooth")
In 2376, Neelix proposed a riddle to Tuvok while the two were on an away mission: an ensign was stranded on an uninhabited moon, without food, and all he had was a calendar. How could he survive? Eventually, Tuvok came to the "illogical" conclusion that the ensign could survive by eating the "Sundays." (VOY: "Riddles")