A small, flat stick, with rounded ends and a flattened tip, used for cleaning the finger nails and pushing back cuticles. It looks like a tiny tongue depressor, about four inches long by half an inch wide (by my poor estimate). They are often orange in color.
One can use an orange stick to clean under ones nails, or to buff and clean ones nails, or to gently push down the cuticle that is growing over the nail. One can do all of these things without a orange stick; in fact, cuticle pushers and nail buffers have been designed to do these jobs more professionally, but orange sticks are cheap to make, easy to carry, and just generally useful.
They are called orange sticks because they were originally made from orange wood. They are, in fact, often still called orangewood sticks. They were invented by Dr. Sitts, an European podiatrist, in 1830. Before this, metal tools and chemicals were used to manicure nails. It wasn't until 1892 that Dr. Sitts' niece introduced proper nail care to women, and the orange stick becomes useful to the everyday woman. The term 'orange stick' was in popular usage by 1910.