The Maya of the Yucatan were a vibrant and sophisticated people. They flourished in the jungles of what is now central and southern Mexico and Guatemala from approximately 1500 BCE to 1500 CE. During their Classical Period (250-900 CE) they created magnificent buildings and monuments to their kings and gods, elaborate works of art, and a sophisticated hieroglyphic language. As with any civilization with grand buildings, maintenance of such an advanced society had its costs on its citizens in the form of taxes and labor, as well as exhaustion of resources. For unknown reasons, but perhaps due to the latter, the Mayan civilization went into decline about the same time Europe was waking from its dark period. In the early 16th century, it was all but stamped out by the Spanish conquistadors.
Living in the thick jungles of Mesoamerica was difficult. The Mayan culture was agricultural, which required accurate predictions of seasonal changes. Their society revolved around a pantheon of gods and the rituals required for them. Intimately tied to their religion was their calendar with its notion of cycles, while central to their astrology was the planet Venus and its cyclical movements. Over time they became expert astronomers and developed a very accurate, though extremely complicated, system of calendrics.
The Maya had an elaborate hieroglyphic written language, with over 700 glyphs, only a portion of which have been translated to date. The vast majority of Mayan writing was on civic monuments or in books made of deer skin, cotton cloth or maguey paper, folded like a fan and now called codices . The primary purpose of these writings was to extol their gods, record the reigns of their kings, commemorate important events, or retell epic battles. Given the often harsh payment in human blood required by their gods, many of the artistic depictions found in the codices are graphic and gory. When the Spanish found these books, they naturally (given their background) viewed them as works of the Devil. Countless codices were destroyed by fire at the hands of mainly one man, Friar Diego de Landa (1524-1579). Sadly or luckily, depending on your outlook, only four intact codices are known to have survived (possibly because they had already been removed from Mesoamerica). The Dresden, Paris, and Madrid Codices are named after the locations of the museum at which they are housed, while the Grolier Codex is named after its New York publisher. In an attempt to save some books, the Maya buried them. But of the ones that have been found, the damp conditions took their toll and only scraps remain . So the vast majority of the Mayan writings and numbers that remain are found on carvings on stelae and buildings, and they include many dates. These dates are given in the context of the Mayan Long Count calendar.
From the cycles of the seasons to the vast cycles of the heavens, the Mayan world view literally revolved around cycles. The Maya believed that the world went through cycles of rebirth and destruction, and that the current world was the fifth such manifestation or "Great Cycle". The current Great Cycle is now over 5000 years old and started on August 12, 3113 BCE in the proleptic Julian calendar, according to Cooke  (though other scholars have calculated a slightly different start date), and will end, possibly cataclysmically, on December 21, 2012. The Long Count gives the time in days since the start of the current Great Cycle. But the Long Count was just one record of time for the Maya, who had at least three other calendars. The Maya calendar system is a marvel of modular arithmetic that accounts for the cyclical nature of the seasons and the motions of the moon, the sun, Venus, and all of the heavens.
Briefly, there were four calendars in use, three for time periods on a human scale and one for the Long Counts. The Tzolkin, which means "count of days," was a 260-day year composed of two independent cycles made up of 13 day-numbers and 20 day-names . The Maya also employed a 365-day solar year known as the Haab or "vague" year because it slowly became out of sync with the seasons. In the solar calendar, the Haab is made up of 18 months of 20 days, plus 5 dangerous "unnamed" days, called the Wayeb. These two calendars align every 52 Haab years or 73 Tzolkin years and together make up a Calendar Round [1, 4]. For much longer periods of time, needed to record historical events, the Long Count was used.
Editor's note: More information about Maya calendars can be found in the following articles here in Convergence.
Maya Calendar Conversions, by Ximena Catepillan and Waclaw Szymanski (2010)
Maya Cycles of Time, by Sandra Monteferrante (2007, reprinted 2012)