A lidded box is a container with a fitted lid made from a mating segment of the same piece of wood. You can see on many of the pieces below that the wood grain aligns quite closely across the join. This is accomplished by minimizing the loss of wood in the forming of the overlapping, fitted portion.
Some of the lidded boxes I turn are made to have what is called a "suction fit." In such boxes, the lid and container fit so closely that a slight resistance is felt in opening and closing the container due to the volume of air that has to flow across the closely mating surfaces.
Click on the images to see them in detail (including species of wood and dimensions of the piece).