What is this game with pegs in holes, jumping and capturing each other?
This is Peg Solitaire, also simply known as 'Solitaire'. The empty hole is usually located in the center of the board, but for variations and different shaped boards this is not always true. This size board makes it relatively easy to simply memorize a winning series of moves, but for larger boards a player may never find that series and will spend more time exploring possible games even from a single starting position.