The cerebellum, Latin for "little brain", is a plum-sized portion of the brain located below the cerebral hemispheres and behind the brain stem. It's main claim to fame is that it contains half the neurons of the brain despite being only 10% of its size. This is because its main constituents are tiny granule cells.
The primary function of the cerebellum is to provide feedback and fine-tuning for motor output. It is also associated with the sense of proprioception, which provides us with an intuitive map of the location of our body parts. Without proprioception, it would be impossible to remain balanced while walking in the dark — we wouldn't have an intuitive sense of where our legs were located. Like many other parts of the brain, the cerebellum was originally associated with a single function, but with the advent of Positron Emission Tomography (PET), fMRI, and other neural imaging techniques, it has been discovered that it is activated in tasks requiring the delegation of attention and the processing of language, music, and other sensory temporal stimuli.
The cerebellum is cytoarchitecturally uniform, like many other parts of the brain. This means that its cells are organized in a very regular pattern, a 3-dimensional network of neural circuits crossing each other perpendicularly. This makes it particularly amenable to staining and study under a microscope. It is therefore appealing for use in instructional laboratory work at universities.
Similar to its larger cousin the cerebrum, the cerebellum is divided into two hemispheres and 10 lobes, all of which have been studied extensively. The cerebrum is one of the phylogenetically oldest portions of the brain. It is very similar across all vertebrates, including fish, reptiles, birds, and mammals. This is strongly suggestive that it performs functions universal to all these species.
Oddly enough, people with damaged cerebellums are capable of leading relatively normal lives. Symptoms of damage to this area include poor motor control, an awkward gait, the overestimation or underestimation of force, and the inability to engage in rapidly alternating movements. Because of the relative simplicity of the cerebellum, attempts at cerebellar modelling are popular among the creators of neural networks and computational neuroscientists.