The bending method pictured is called bent lamination. You take multiple thin sheets of wood and glue them together in a bending form, as illustrated in https://woodworking.stackexchange.com/a/1795/49.
- Use single veneers. These are thin sheets of wood, often around 1/40"-1/8", though they can be thicker or thinner. Obviously the thinner the veneers, the more you'll need to glue together to get to your final thickness. You can buy veneer or resaw your own veneers on a bandsaw. If you're fortunate enough to have access to a drum sander, it will guarantee that each veneer is of uniform thickness throughout, though it is not necessarily important for all veneers to be the same thickness as each other.
- Use bending plywood. This plywood is not only thin, but it has the grain of all the plies oriented in the same direction rather than criss-crossing the grain of alternating layers.
- If the bend is not very severe, you can also use thin ordinary sheets of plywood (even if they are not technically "bending plywood").