There are a number of theories that claim that the world will end in December of next year. They have sprung up, generally speaking, because of the calendar system used by the Maya.
The Maya's calendar system was based on three different calendars. There was a sacred calendar of 260 days. There was a solar calendar that had 365 days. Finally, there was a "Long Count." This was based on the idea of 52 year cycles because the other two calendars combine to make a cycle that repeats every 52 years. Included in this idea of the long count was the idea that they were living in a period that had begun in roughly 3100 BC and would go for about 5125 years. That period runs out next year.
Some people say that this meant that the Maya thought the world would end when that period ran out. Others (see National Geographic link below) argue that this is not that much different than the changing from the 1990s to the 2000s -- it's just the start of a new cycle.
People now worry that the world will end next year because of this, but it seems kind of strange because by this calendar, the world only began in 3100 BC. So I don't really see why people believe that the world will end by this Long Count when it clearly didn't begin when the Long Count said it did.
The Mayan Calendar like all calendars indicates time periods but does not predict anything. In fact, more recent research by the Dutch shows there were erros in matching the Mayan Calendar with our modern calendar. When those errors are corrected it is found that the Mayan Calendar, instead of ending in 2012 actually ends in 2220.
Here is the original Dutch Article
and here is the google translation
the one thing that also makes people think that the world will end in 2012 is the movie 2012 which is a movie based on a calender.