The corpus callosum, a big white fibrous band, winds back and forth between the left and right hemispheres, thus allowing each hemisphere to communicate with the other. A particularly interesting feature of the corpus callosum is what happens to the brain when it is severed.
In fact, studying what happens to people who have had their corpus callosum severed has allowed neuroscientists to isolate each hemisphere and study their functions and how they operate independent of one another.
Marilee Sprenger shares a short tale of how the two hemispheres operate (page 42).
You meet a person for the first time. After a short conversation you part company. As you walk away you think about what your new acquaintance said. You decide that this could be your new best friend. Your left hemisphere analyzed the words spoken. Your right hemisphere analyzed the tonality, tempo, volume, body language, and perhaps the attitude displayed. The two sides agreed that this was someone with possibilities as a future friend.
You can see the corpus callosum more clearly at this Indiana University site.