The brain is only a part of a larger nervous system that extends to each part of the body. The nervous system can be divided into central nervous system (CNS), which consists of the brain and the spinal cord, and peripheral nervous system.
Superior to the pons, a structure named tectum contains inferior and superior colliculi that are highly important structures for early processing (and also in the case of superior colliculus fusion of ) auditory and visual information, respectively. Another structure that is located superior to the pons is the tegmentum. The ventral tegmental area has been intensively studied as the source of specific neurochemical transmitter substance dopamine that has been associated with reward circuitry of the brain that is important for motivation and is also implicated in development of drug addictions.
Cerebellum is located posterior to the pons. The cerebellum is a structure that has been implicated in coordination/timing of well-learned motor sequences, such as a golf swing executed by an experienced golfer. Damage to the cerebellum results in deficits in fine motor coordination, balance, and motor learning. The cerebellum gets inputs from sensory and motor cortical areas of the brain and sends output through the so-called deep nuclei to thalamus and cortical (especially premotor and motor) areas.
Recently, there has been increasing recognition that the cerebellum, in addition to its role in motor coordination, is important for cognitive functions such as selective attention and emotions (Schmahmann, 2010). The posterior lobe of the hippocampus is especially implicated in other than motor functions. Thalamus, on the other hand, is a very important structure through which most of the sensory inputs travel to cortex (as well as being a major target of cortical descending inputs) and hypothalamus regulates hormonal secretion via the pituitary.
To summarize, the evolutionarily older structures that underlie the cerebral hemispheres are mainly responsible for regulating vital body functions, arousal, and relaying ascending inputs from the senses (and also descending motor commands). If humans only had these structures, we would however, lack what makes us human; the ability to reason, remember, plan ahead, feel for others, be social, possess sense of humor etc. All these are made possible by the cerebrum that is the part of the brain that has expanded the most during evolution.