The mainstay of a neurologist's work is diagnosing disorders of the nervous system...the brain, the spinal cord, and peripheral nerves. The disorders include diseases such as epilepsy, Alzheimer's, Parkinson's, Lou Gherig's disease, Multiple Sclerosis, etc., as well as damage done from trauma, vascular problems (stroke, etc.), and toxic/metabolic disease.
Neurologists typically do not do any interventional things, i.e., they don't do surgery. They do read and interpret MRI, CT Scans, and X-rays, as well as specialized neurologic tests such EEG's and EMG's (EMG's test nerve conduction and the neuro-muscular junction). They tend to be excellent at taking a patient history and doing a thorough physical exam to determine a diagnosis.
Some difficulties in the field of neurology include the lack of adequate treatment for many of the disorders, and that neurologists are often required to be the bearer of bad news (brain tumor, little brain activity, terminal and untreatable disease, etc.). Compensation in neurology is not as high as other fields in which there are more interventional treatments.
Residency is four years long. First year is typically completed in internal medicine in which the hours are long and grueling. You will probably be on call every 3 days (which means you should be prepared to stay awake for 24 hours every third day although this won't always happen...by the way this is true for just about every residency for at least the first year). Hours get better and better as the residency goes along.