Parkinson's disease is a degenerative disorder that involves a specific set of dark neurons in the midbrain called the substantia nigra . These neurons are dopaminergic, meaning that they produce and use the neurotransmitter dopamine. They project to many parts of the brain, but the important projection in terms of Parkinson's disease goes to the basal ganglia, a set of brain areas that help to control movement. The basal ganglia consists of two areas, the striatum and the globus pallidus, as seen in the diagram below. The lack of a dopamine input into the striatum, results in an increase in the output of the globus pallidus to the thalamus. This is an inhibitory input, meaning that the result is the rigidity and awkwardness of movement that one sees in people suffering from Parkinson's. Presently there are treatments that help ease the symptoms of Parkinson's, such as L-dopa, but none that can offer a long term cure. This is because it they do not replace the lack of input from the substantia nigra. Though neural stem cells do not offer a total cure, they can offer more improvement and a long term gain in functionality.
The treatments that are currently being developed, involve placing a graft of stem cells into the striatum. This graft needs to be already differentiated into cells that are likely to become ones which secrete dopamine (technical information on cell differentiation). Once these cells are differentiated they are placed on the striatum, where they provide the proper amount of dopamine.
You may be wondering why the graft was done in the striatum and not in the substantia nigra where the original neurons are lost, and why it works if the graft is in the wrong place. This is because it easier to make a graft in the striatum. If it was put in the substantia nigra the projection axons would have to stretch all the way from the midbrain to the basal ganglia. In animal models it appears that the striatal graft is successful in relieving parkinsonian symptoms. This may be because some of the dopamine released by the substantia nigra is done in a constant fashion, and the graft could simulate this influence. It also appears that the graft receives information from the cortex and other thalamic areas, and thus the brain may be flexible enough to accommodate for a new source of dopaminergic input to the basal ganglia (Bjorklund and Lindvall 2000 picture also).

Human trials of this method have been fairly successful. Curt Freed, et al (and picture below) conducted a study which used a control group of patients who received a fake surgery and did not receive the stem cell graft. This was considered neccasary because Parkinson's disease can become better through the power of the placebo effect. That is when the patient believes he is receiving a treatment, but is not really getting one. In this controlled study, they found that patients who received the stem cell graft had improved scores on motor tasks. However, this seemed to only work with patients who were younger than 60 or who showed a good response to the drug L-dopa. This may be because when the disease damages to much of the brain, the graft will not work.
Unfortunately, one of the patients died due to heart failure. Examining his brain after his death, the scientists found that the neural stem cell graft had worked well. The cells had differentiated into the proper dopaminergic neurons and were making and receiving connections with other neurons. The picture below is of one of these neurons. Notice the long projections it makes, indicating that it is connected to many other neurons.

see the Parkinson's disease web site in The Undergraduate Behavioral Neuroscience Resource Project.