The wood is walnut. Work wood glue under the lifted veneer and clamp it down flat overnight until the glue dries. You can buy walnut veneer and fill in the missing pieces. Try to match the grain as much as possible and trim the veneer to fit. Somewhere like Rockler will have it or buy some on ebay. Just do a search for walnut veneer. There's some pretty serious damage but to me the cabinet and stool are worth the effort. Sanding it down without replacing the missing veneer will remove more veneer and expose the much less expensive and plainer wood of the substrate.
Rodney