In short, no, Mayan people did not predict that the world would end on December 21, 2012; this belief is not only a misinterpretation of Mayan culture and timekeeping but also, if taken literally, inaccurate from the perspective of the Mayans. The Gregorian calendar that includes dates like December 21 was not developed until the 16th century C.E., so even if they had predicted the end of the world, the Mayans wouldn't have said that December 21, 2012 would be the day. Additionally, the very notion that Mayans predicted the end of the world or thought that time would cease to exist at a certain point is based on an incorrect interpretation of the way this culture kept time.
The date that corresponded with December 21, 2012 on the Mayan calendar was indeed the end of something, but that ending was not so dramatic as to include the existence of life. Instead, the Mayan calendar was based on 400-year cycles known as Bak'tun. The December 2012 date was simply the end of one of those Bak'tun. If the Mayan civilization still existed in 2012, they would simply have celebrated the end of an era and rolled the calendar over to start anew.