Thirteen.
How old

It must be very difficult to stop younger players coming to the site - but I'm fairly sure the site says you must be at least thirteen to play here.

Yup: http://www.chess.com/legal.html
Chess.com is concerned about the safety and privacy of all its users, particularly children. For this reason, parents or guardians of children under the age of 13 who wish to allow their children access to the Service must create an account that is maintained by a legal guardian of at least 18 years old and monitor the child's usage. By allowing a child access to your account, you also give your child permission to access all areas of the Service. Please remember that the Service is designed to appeal to a broad audience. Accordingly, as the legal guardian, it is your responsibility to determine whether any of the Service areas and/or Content are appropriate for your child.
How old do you have to be to be on this site?