Each of the three Mayan calendar cycles had a day zero, which was the beginning and end of the cycle. Normally, when one cycle came to an end, at least one of the others was still in mid-arc. However, very rarely, the three zero days would coincide.
The Mayans took their calendar very seriously - the cycles of the calendar determined the seasons, harvest time, the days that the gods demanded sacrifice, and everything else. They were also the first people of the Mesoamericas to come up with the concept of "zero," and they found it terrifying - they worshipped the God of Zero as one of the main gods of the underworld. This coincidence of zero days, the ultimate End to All Cycles, was thus an occasion of great dread to them. What, they wondered, will ensure the continuation of time, when everything has come full-circle and returned to Zero?
The only answer was to placate the gods, which - typically for the Mayans - meant sacrifice. The victim was dressed in the regalia of the God of Zero, and worshipped throughout the great festivals held in the gods' honor. At the climax and end of the celebration, the priests sacrificed the victim by ritually tearing off his lower jaw - perhaps symbolically removing the "teeth" of the awful Zero. The gods were then placated, and all things - demarcated by the endless turning of the cycles - could resume again.
Source: Robert Kaplan's The Nothing That Is: A Natural History of Zero