The Long Count calendar used by the Mayan people in pre-Columbian Mesoamerica "turns over" in the year 2012. Some have taken this to mean that the Mayans believed that the world would end on that date. However, there is no evidence that they actually held such a belief.
Many ancient civilizations had their own unique calendar systems, but only a very few remain in use in modern times. Some we know of only from archeological evidence, and in many cases we may not know much about them. There may have been many civilizations that made doomsday predictions based on their calendars, predictions that we no longer know about or cannot interpret because our knowledge is incomplete.
The Mayan calendar, however, is one ancient calendar system that we are fortunate enough to have fairly good information about. In fact, they effectively had two calendars that they used simultaneously: the Calendar Round and the Long Count. It is only the Long Count that seems to concern those who believe in the prophecies of doom.
The Calendar Round consisted of a 365-day cycle and a 260-day cycle. The Haab' cycle consisted of 18 months of 20 days each, followed by 5 intercalary days that were considered unlucky. The 260-day cycle was a little different from what we're used to in the modern world; it combined a repeating sequence of 13 numbers with another repeating sequence of 20 day names to produce a pattern that repeated every 260 days. Each day would be labeled with a Haab' month and day number (or be one of the Haab' cycle's 5 intercalary days), along with a day number and name from the 260-day cycle. A little math will tell you that this scheme will repeat every 52 Haab' "years." This article won't go into more detail about the Calendar Round, but as you can see, one of the two calendars the Mayans used was fairly complex.
The other calendar, the Long Count, was strictly numerical. The Long Count used special units of time measurement, all based on k'in (which means "day").
1 k'in = 1 day
1 winal = 20 k'in (20 days)
1 tun = 18 winal (360 days)
1 k'atun = 20 tun (7200 days)
1 b'ak'tun = 20 k'atun (144,000 days)
There were also longer units, but they were not used as frequently, so we have less information about their lengths.
Long Count dates are represented by a sequence of numbers, starting with the number of b'ak'tun and continuing to list the number of k'atun, tun, winal, and k'in. The Mayans and other Mesoamerican groups did not use the Arabic numerals we commonly use today, of course, but when talking about Long Count dates now, we mostly just list the numbers separated by a period. For example, 18.104.22.168.5 would mean 1 b'ak'tun, 2 k'atun, 3 tun, 4 winal and 5 k'in (or 159,565 days) after date 0.0.0.0.0. But this brings up the question of how to correlate the dates with the Gregorian calendar that most of the world uses today. What is today's date in the Mayan calendar?
Most evidence seems to point to a date of August 11, 3114 BCE as the start date for the Long Count (corresponding to 0.0.0.0.0), although there is no evidence to suggest that the Long Count was actually used so long ago. However, this places 22.214.171.124.0 on September 18, 1618 CE, and it places 126.96.36.199.0 on December 21, 2012.
Mayan legends say that the gods have created the world four times, the first three attempts having failed in one way or another. Each time, the gods would wipe the slate clean and start again. The previous creation ended, according to legend, with the start of a 13th b'ak'tun in the Long Count. There is no mention of when the first two creations ended. But some modern-day readers have used this to imply the existence of some sort of prophecy that the current fourth creation will also end with a 13th b'ak'tun.
The Mayans themselves do not seem to have believed that the world would end on such a date. Keep in mind that there are larger time units than the b'ak'tun, and although Long Count dates were usually abbreviated to just the last 5 numbers, there are date inscriptions that include the larger units, some of them mentioning dates farther in the future than 188.8.131.52.0. It seems likely that they would consider this date auspicious, perhaps a sign that a significant event would occur or an occasion for a great celebration, but it does not seem as if they expected the world to end then.
To summarize, there is no prophecy that the world will end with a 13th b'ak'tun, there are Mayan references to dates after 184.108.40.206.0, it is not certain precisely what date corresponds to 220.127.116.11.0 in modern terms, and we may be placing more importance on the Mayan calendar than it warrants merely because we have more knowledge about it.