The human brain is physically divided into two hemispheres that connected to one another through the corpus callosum which is a fibrous band of tissue that transmits information to and from the two hemispheres.
These two hemispheres, the left and the right, are capable of working independently as having been proven by procedures in which it was necessary to physically sever the corpus callosum and separate the two halves. Hemispheric specialization refers to which specific cognitive functions are controlled by which hemisphere of the brain.
For example, the left hemisphere of the brain is responsible for language functions such as production of speech and language comprehension. The right hemisphere has more visual-spatial functions. Hemispheric specialization is the basis of the "right-brain, left brain" personality differentiation that is popularly referred to in "pop" psychology (which is not a scientifically valid concept).