The Mayan Long Count calendar rolls from Baktun 12 to Baktun 13 on that day, but it’s not an ending. It goes up to Baktun 19 and then Pictun 1 begins (A Pictun is 20 Baktuns; a Baktun is about 400 years).
For some reason people think the Long Count calendar is very mystical. It’s not. It’s very simple. It’s like an odometer, only in most spots it counts to 20 instead of 10. It was not a religious calendar.
The calendar doesn’t predict anything; it just counts days since the beginning of the world (according to the Mayans, on September 7, 3114 BCE). Not sure who the "they" is who "say the world will end" that you reference in your question. Modern Mayans don’t use the Long Count anymore; they only use the 260-day sacred calendar--that was/is the religious calendar.
The general belief of the Aztecs and Mayans was that the current world would end during a 5 day period in their other civil calendar (the 365-day one). Right now this five day period falls every year at the beginning of April (moves 1 day each leap year)--not anywhere near December 21. The Aztecs in specific believed that the last day would be 4 Earthquake (which comes around every 260 days in the sacred calendar)--12-21-2012 is 4 Flower.