A number of different woods can be used to make a Chess board, the overriding factor should be that the light wood should be a light as possible the the opposite for the dark wood. It takes something from the effect to have a board made with dark wood for whiteand darker wood for black. This is particularly true when the "real" boards used in tournaments are black and white, literally. Additionally, the squares are called black and white, too. Contrast is important. Certainly you are less limited when making a board out of wood today because of the large number of darker stains and wood lighteners available to treat wood.
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Most chest sets are made from wood these days. However, if you get a more expensive chess set it could be made out of many different minerals and some even out of gold. The cheaper, more common ones, however, are made mostly from wood.
There are many materials they can be made out of. It's basically your choice. They make a lot of fancy chess sets. A few examples; Plastic, metal, glass and stone.