The term used to denote fear of Friday the 13th is called paraskavedekatriaphobia, a specialized form of triskaidekaphobia, a phobia (fear) of the number thirteen.

Apart from that, and the slasher film, we all have that quirky feeling when such a day comes around.

Quoting from some random, strangely irrelevant internet source ...

The end of the 13th b'ak'tun (December 12, 2012) is conjectured to have been of great significance to the Maya, but does not necessarily mark the end of the world according to their beliefs, but a time of re-birth. According to the Popol Vuh, a book compiling details of creation accounts known to the Quiché Maya of the colonial-era highlands, we are living in the fourth world. The Popol Vuh describes the first three creations that the gods failed in making and the creation of the successful fourth world where men were placed. The Maya believed that the fourth world would end in catastrophe and the fifth and final world would be created that would signal the end of mankind.

In reality, the Mayans believed that the world would end on October 13, 4772, which is a Friday. This is confirmed by a date from Palenque, which projects forward in time to The Classic Period Maya obviously did not believe that the end of this age would occur in 2012, but on this later date.