14th January
The first thing you need to know is that the Mayans did their arithmetic in base 20, not base 10. So they had individual symbols for each number from 1 to 19. And they stacked the digits above each other rather than to the left. See examples at right.

Now for the calendar. Two calendars, actually. One is for working out when to carry out various rituals. There are 20 day names and thirteen numbers. You start with name one, number 1, and so on up to name thirteen, number 13. The next day is name fourteen, number 1, then it continues to name twenty, number 7, then name one, number 8, and so on. This process continues for 260 days before a name-number combination repeats, and then the whole 260 day process repeats again.

Meanwhile, there's another, 365 day calendar, presumably more useful to people planning things like harvests. This has 18 months of 20 days, with a special 5 day 'loose' period at the end of the year when all sorts of terrible things might happen.

So, all told, each day has four identifiers: a day name and number from the ritual calendar, and a day number within a month from the 365 day calendar. How often for any particular combination of the four to come around again: 52 years. This cycle is called a calendar round, and is a highly significant period in much Mayan mythology.

However, this still wasn't satisfactory to the Mayan astronomers, who wanted some way to distinguish days that happened to be exactly 52 years apart. Hence the "long count calendar". 20 days in a uinal. 18 uinals in a tun. 20 tuns in a katun. 20 katuns in a baktun. 20 baktuns in a pictun. 20 pictuns in a calabtun. A calabtun turns out to be around 160,000 years: why would they feel the need for a word to describe a period of that size? The mind boggles.

Two other comments: the Long calendar started on 13 August, 3114BC. No-one knows why. A period of great significance to the Mayans is a great cycle, of 13 baktuns - around 5,200 years. The current great cycle thus ends on 23 December, 2012, when presumably something really important will happen.

Finally, a quick word on astronomical whatnot. The windows of the observatory at Chichén Itzá are cunningly aligned to let you spot equinoxes, solstices, and other less obvious stuff like the most northerly and southerly points in the movement of the planet Venus. Other windows appear to be lined up with the movements of other starts, particularly those of the zodiacal symbols which lie in the plane of the ecliptic. Not to mention the Castillo at Chichén Itzá, which is designed so that twice a year the shadows on one of the staircases look like a moving serpent, an effect enhanced by the serpents head built into the bottom of the staircase, producing a show that lasts for over 3 hours. The fact that this effect reaches it's most convincing aspect at the equinoxes is a remarkable tribute to both the creativity and planning skills of the architects of the day.