People mean different things when they say that a part of their body feels "numb."
Sometimes they mean that they can't feel anything at all, almost as if the body part weren't there. If the body part were to be jabbed with a fork, they wouldn't know.
Other people use the word "numb" to mean a tingling sensation, like their arm has fallen asleep. The classic description is a "pins and needles" feeling. Others mean a kind of shooting electric sensation. These are examples of paresthesias-- feelings in the body that shouldn't be there.
Transient paresthesias are extremely common. It may be difficult to imagine someone who has not experienced this sensation. They can be caused by lying on a limb, or by breathing too fast. They usually resolve quickly.
Paresthesia that doesn't go away is another matter. Chronic paresthesia is a signal that something has gone wrong with the pathways that normally allow for sensation. It is often related to the peripheral nervous system. For example, in vascular disease, the peripheral nerves may not get enough blood, and as a result send unusual messages to the brain that are interpreted as paresthesias.
To learn more about tingling and numbness, start here: Tingling and Numbness.