Benjamin Libet conducted this experiment in the 1970s. Apart from one or two electrodes on the scalp, there's really nothing creepy about the experiment. Until you read about the results.
Libet asked his experimental subjects to move one hand at an arbitrary moment decided by them, and to report when they made the decision (they timed the decision by noticing the position of a dot circling a clock face). At the same time the electrical activity of their brain was monitored. Now it had already been established by much earlier research that consciously-chosen actions are preceded by a pattern of activity known as a Readiness Potential (or RP). The surprising result was that the reported time of each decision was consistently a short period (some tenths of a second) after the RP appeared.
The RP starts to ramp up as much as 0.3 seconds before the reported decision time. It continues to increase after that, leading up to actual hand movement about 0.2 seconds later.
What does this mean? All the test subjects, of course, felt they had consciously chosen to move. But if unconscious brain activity precedes the conscious experience of decision-making, then surely we must conclude that the decision is not consciously made. Effects don't precede causes.
Now there are countless philosophical objections to this conclusion. Some philosophers claim that to interpret this result at all competently, you have to be well-versed in the philosophy of the mind. Which seems reasonable enough, but it's a deep field with centuries of literature in many languages. So this prerequisite rules out anyone who has spent his life studying anything as patently irrelevant as mere neurology. To say nothing of random bloggers.
I'll wade in anyway, of course. Just don't get the impression I know anything about this subject. I don't.
There's a really nice, compelling interpretation that permits free will. It goes like this. The way the mind interprets time is anything but objective. In the Libet experiment, what's happening is that the mind shifts the experience of deciding to move forward in time and shifts the experience of motion backward in time, effectively bringing them closer together. So the conscious decision to act actually does cause the RP ramp-up. But the subject incorrectly reports the decision as having happened later, because his brain has deceived him about the timing.
Why would the brain do this? It seems likely to me that there's an evolutionary benefit to perceiving decision, action, and effect as a single event. I don't think we're equipped to deal with that kind of time lag consciously. Just think—there's a half-second lag between when you decide to move a muscle and when it moves. Have you ever played the piano? If you were aware of this lag all the time, could you do that? Could you run? Could you fight?
An article, “Free Will and Free Won't” (in American Scientist; $12 to download the PDF from their site), puts the Libet experiment alongside four or five other rather clever experiments into will and consciousness. Then it starts talking about alien hand syndrome. The brain is strange.