What is this Old French Game board with 10x10 squares and dark and light circular pieces?
That appears to be a relatively nice double sided International draughts/Checkers board.

Without any further identifying information, it could have been produced anytime in the last 1-200 years, but based on its appearance, I'd estimate it was manufactured around the 1940s-1950s.

The markings indicate the board has a Breveté SGDG, which is a French patent, the Breveté SGDG system started in 1844 and ended in 1968, so that narrows the age down to a 120 year window.