It depends on what you want to clean. The wooden furniture portion of the piano, including high-polished finishes, is usually cleaned with a soft, lintless cloth, slightly dampened with water if you wish. If the wood has a grain, wipe in the direction of the grain. Be careful about using commercially available furniture polish; manufacturers sometimes recommend against it. There are special polishes available from piano technicians or online for different kinds of piano finishes. In any case, follow the piano manufacturer’s recommendations if available. A good article about cleaning and polishing a piano’s finish can be found here.
For cleaning the key tops, slightly dampen a cloth with a mild solution of soap and water and rub the key tops, but do not let any water run down the sides of the keys. Then dry the key tops off right away with a dry cloth. Especially if the key tops are made of ivory, do not let water stand on them for any length of time, or else the ivory may buckle and come loose. Rarely, stain from the black keys will rub off, so you may wish to use separate cloths for black and white keys.
To avoid causing damage to delicate parts, we recommend letting a piano technician clean the soundboard under the strings, the inside of the action, and under the keys.