What are the rules for playing with this unusual Mancala board?
Sounds like you've got a Bao board. 



Bao, an East African Mancala, is the only known Mancala to make use of square pits as well as round ones. These are called nyumba (house), one for each player, and marks your starting position. There are also some extra rules for the nyumba in the main variation Bao la kiswahili. The Wikipedia article I've linked to has detailed instructions for both simplified (Bao la kujifunza) and Bao la kiswahili. 

You can also use the same board, and ignore the shape of the special pits, to play other four-rank Mancala variants. One example would be Omweso. Enjoy the cool board!