Solitaire played with marbles is called "Solo Noble" or "peg solitaire." It requires a board with either 33 or 37 holes designed to hold marbles, depending on the version being played.
The English version of peg solitaire uses a board with 33 marble holes arranged in a cross pattern, while the European variant of the game requires 37 holes arranged in an octagonal shape.
In both cases, however, the basic rules of play are the same. Each of the holes are filled with marbles, with the exception of the hole right in the center of the board. Marbles are then jumped over each other into an empty hole, in a manner similar to checkers, except that jumps are only allowed in the four cardinal directions. Marbles that have been jumped over are removed from the board, and the objective is to clear the board of marbles. Of course, one marble is necessarily left, and this marble must land in the center hole with its final jump.
Since the board always begins in the same state, once a solution is found it can always be repeated in future games. However, there are multiple possible solutions for each type of peg solitaire, adding replay value for those who have already found one of the solutions.