One experiment you have probably heard of is that letting amputees "see" a lost arm or leg with a mirror that doubles the remaining one allows them to scratch or move it. That is, scratching the reflection they see in place of the lost body part does register in the brain, even though there is no direct sensory input. Some months back, we also learned about the "body swap" illusion that makes use of somewhat more sophisticated technology to create the illusion that one is moving a different body, with the aim to test how readily the brain accepts it as one's own. The SciAm Mind article suggests some low tech experiments you can try at home. For example, using a mirror to produce an image of your hand in place of the actual hand and then stroking the image produces a conflict in the brain because the visual input doesn't match the expectation. As a result your hidden arm might feel numb, though there's nothing wrong with it.
This reminded me of a trick we used to play on the mind as children: Lock hands with a friend, with the index fingers straight (see image below). With the free hand, rub up and down your and your friend's index fingers (2nd image). We used to call it "rubber finger." Everybody I know who tried found it to feel weird. I don't know why, but it seems that the brain expects some signal from the friend's finger. It doesn't make a lot of sense to me since you'd need three hands for that. If you have a good interpretation, let me know.