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," though the original answer was that he had "feasted on the dates." (VOY: "Riddles")