A living hinge is a flexible hinge made from the same parent material it connects. There are many designs you can use to allow an otherwise rigid material to flex, enabling you to create decorative designs.
For the most part, laser cut objects consist of flat panels that are either cut or etched before being assembled into a final configuration. There is another way to use laser cutting to create 3D objects from flat materials, and it uses a clever design element called a Lattice or Living Hinge.
MDF and PLY are great to create a living hinge using a laser cutter because the properties of these materials allow for a significant degree of flex before it fractures. Acrylic sheets can also be used. Hinges work by exploiting the natural tendency for materials to flex around an array of cuts or notches.