The "theory" of left/right brain dominance originated from the work of Roger W. Sperry. In order to reduce or eliminate seizures, epileptic patients had their corpus collosum severed (the structure that connects both hemispheres of the brain). As a result, his patients experienced a sort of "split mind", where they were able to name objects presented to the left side of the brain, but unable to identify objects presented to the right side of the brain. He concluded that the left side of the brain is responsible for language.
From this research evolved the pop psychology phenomenon of left/right brain dominance "theory". I say "theory" because it is not supported by empiricism, nor has it ever been seriously considered by neurologists as having any empirical weight. While the left side of the brain houses much of the areas responsible for language, and the right side of the brain houses much of the areas responsible for advanced visual processing, it is not as simple as that.