The right hemisphere of the brain operates the muscles in the left side of the body and the left hemisphere of the brain operates the muscles in the right side of the body as shown in the labeled picture on the left. Damage to a certain side of the brain effects the way one side of the body works that it corresponds to.

Sources:
http://faculty.washington.edu/chudler/split.html
http://capone.mtsu.edu/studskl/hd/hemis.html
http://www.livescience.com/32523-why-are-some-people-ambidextrous.html