What is this board game with black & white pieces and criss-crossing lines called? or how is it played?
This is a game called "Fighting Serpents" or "Kolowis Awithlaknannai",
and in particular this is a version of that game produced by a Greek company by the name of Elite Crafters.

The game originally comes from the Zuni people of New Mexico, but its most likely a modified version of Alquerque, which was introduced by the Spanish while they were exploring the Americas.