What does one actually "do?"
One generally sees patients. You diagnose, evaluate and treat. Usually this involves both inpatients and outpatients, although depending on the practice there may be more or less of one or the other. Like internal medicine, neurology is increasingly using hospitalists to manage inpatients. If you are in an academic setting you may also teach and/or do research.
You may or may not also do diagnostic testing such as lumbar punctures, EEG, EMG, or sleep studies. A few neurologists are trained in interventional neurovascular techniques.
There are many subspecialties with in neurology, so you may end up seeing mostly stroke, mostly epilepsy, mostly movement disorders etc depending on your interests. The vast majority of neurologists, however, do still see at least some random "general neurology" patients who may be outside their subspecialty interest.
Lifestyle depends on practice environment. In private practice, the surprising truth is that you can generally work as much or as little as you want to, but your income will directly reflect the amount you work (i.e., no work = no money coming in from insurance companies = no pay). Income in private practice for neurology can range from under $100K to well over $300K, but at that level you will be super-super-busy. If you are directly employed by a hospital or HMO, you will likely lose this freedom to set your own workload, but generally have a "guaranteed" level of salary.
Also, please describe the typical outside-of-work lifestyle (family time, on/off-call time, etc.).
Family time depends on how much time you choose/need to devote to work. Again, can be highly variable, especially in private practice. Just remember that whenever you're not at work, you're not making money . . .
Call is highly variable. If you are in a large group practice you may only be on call once or twice a month. If you are the only neurologist for 100 miles, your call may be pretty frequent. Also, being called at night doesn't necessarily mean you have to run in and see the patient in the ER (unlike residency, where that's required). If you're affiliated with a group that has hospitalist physicians or a really strong ER, you may just discuss the case over the phone and then actually go see the patient the next day.
Also, how does the future look for neurologists? Are there any major reimbursement reforms, etc. on the horizon for neurology?
Looks great for neurologists because neurologic disease clearly will increase with the aging of the population. Particular areas that will really take off are stroke, dementia, movement disorders, and sleep.
The increasing need does not, however, necessarily correlate with increasing income, which goes to the point of your last question. There are always going to be reimbursement changes. Almost always they will be in the downward direction (although there are exceptions. for example, the reimbursement for hospital stroke care was recently increased). The major income generators in neurology right now are EMG and sleep studies; if there are big decreases in reimbursement for those, neuro will take a major hit.