The right side of the brain controls muscles on the left side of the body and the left side of the brain controls muscles on the right side of the body. Also, in general, sensory information from the left side of the body crosses over to the right side of the brain and information from the right side of the body crosses over to the left side of the brain. Therefore, damage to one side of the brain will affect the opposite side of the body.
In 95% of right-handers, the left side of the brain is dominant for language. Even in 60-70% of left-handers, the left side of brain is used for language. Back in the 1860s and 1870s, two neurologists (Paul Broca and Karl Wernicke) observed that people who had damage to a particular area on the left side of the brain had speech and language problems. People with damage to these areas on the right side usually did not have any language problems. The two language areas of the brain that are important for language now bear their names: Broca's area and Wernicke's area.
Broca's Area
Wernicke's Area
Each hemisphere of the brain is dominant for other behaviors. For example, it appears that the right brain is dominant for spatial abilities, face recognition, visual imagery and music. The left brain may be more dominant for calculations, math and logical abilities. Of course, these are generalizations and in normal people, the two hemispheres work together, are connected, and share information through the corpus callosum. Much of what we know about the right and left hemispheres comes from studies in people who have had the corpus callosum split - this surgical operation isolates most of the right hemisphere from the left hemisphere. This type of surgery is performed in patients suffering from epilepsy. The corpus callosum is cut to prevent the spread of the "epileptic seizure" from one hemisphere to the other.
Left Hemisphere
Language
Math
Logic
Orderly
Critical
Analytic
Scientific
Right Hemisphere
Spatial abilities
Face recognition
Visual imagery
Music
Random
Expressive
Emotional
Intuitive
Responsive
Source: http://faculty.washington.edu/chudler/split.html