The two hemispheres of the brain control two different parts of the body because the nerves that travel down from the brain to the periphery (hands for example) cross over at the medulla (pyramid decussations to be specific). This means on the Left hemisphere of the brain in the motor cortex the nerves will travel down ipsilaterally (same side) UNTIL they reach the medulla. At that point they cross over so the original left hemisphere axons will be on the right side of the body (contralaterally). They continue on this other side to the spinal cord where they will send the message further to the target.
This nerve pathway is called the lateral corticospinal tract. Hope this helps!