The ability to move your right hand is controlled by the

Left Hemisphere of the cerebrum

The brain (cerebrum) is composed of the left and right cerebral hemispheres which are separated by a fissure and joined by a large bundle of nerve fibers known as corpus callosum. The left hemisphere of the cerebrum is the logical side of the brain and it is responsible for the control of the right side of the body while the left side of the body is controlled by the right cerebral hemisphere.