The game's called Go, and the rules are actually fairly straightforward (the strategies are where it gets ludicrous), and there's a lot of similarities to what they've been playing. You place the stones on where the grid lines intersect, and the goal here is to surround more territory than your opponent. Stones in a horizontal or vertical line are treated as one stone. Surrounding an enemy stone so it has no adjacent free spaces means you capture it. You get points for intersections under your control and for stones captured.
You can't make a move that essentially resets the board - like, if Tetsuma captures Dayo's piece, Dayo can't immediately capture the piece Tetsuma used by putting his piece back where it was (Tetsuma demonstrates what this would look like). Though you can have two situations set up like that and get stuck in an eternal loop.
The game doesn't have a set end; instead it goes until they decide to call it quits.