Nearly every village in Mexico has a church and a market the city's public square where the market is held is called a?

This square is generally called plaza, or in some cities specifically, Zocalo. Zocalo refers to the base of the flag - in many such squares there is a (relatively) big flag, and the squares are also referred to as Zocalo, after the name for the base on which the flag is positioned. The most famous Zocalo is in Mexico City.