A medium is a parapsychological professional dedicated to proving the existence of the persistence of human consciousness after death, by communicating with the spirits of the deceased. This communication usually comes about in the form of some sort of sensation or sensory input that only the medium can detect, whether it's clairvoyance (visions of the departed), clairaudience (hearing messages from the departed) or clairsentience (where they detect the emotions and thoughts of the deceased).
Mediums got their start in Western culture with the rise of Spiritualism in the 1840s, first in America and then crossing the Atlantic to Britain. Mediums became part of the conventional grieving process, and a way for bereaved family members to gain some semblance of closure with their departed relatives. In African traditions, the lungam or voudoum intercedes for spirits in the material world, appeasing them, or directing their still living relatives to correct wrongs. In Asian traditions, spiritualism ties directly to ancestor worship.
As a source of controversy, mediums have been unparalleled. Harry Houdini, the great escape artist, spent the last part of his life attempting to debunk spiritualism and any kind of communication with the afterlife, and spent considerable sums of money in doing so. His crusade on this subject came about because of what he considered charlatans using psychology and mummers tricks to attempt to convince him that he'd communicated with his deceased (and beloved) mother.
His willingness to use his knowledge of stagecraft and illusion led him to being part of the panel of experts for Scientific American that offered a prize, as yet uncollected, for incontrovertible proof of an afterlife and communication with it. This tradition continues to this day with notable skeptics James Randi and Penn & Teller making part of their act debunking claims of the supernatural.
The attempts at debunking mediums have not put a damper on their business; there are many mediums out there, practicing their craft, from the famous (such as American John Edwards) to the relatively obscure (Myrah Blackledge). All of them claim to be entertainers, and to some extent, grief counselors. Their shows usually revolve around finding a person in the audience who shoes clear signs of bereavement, then asking carefully structured questions, where they can 'fish' for answers that get notable responses in body language, posture and facial expressions.
Some claim that this is fraud of the most exploitative sort, preying on the emotions of people who are grieving. Others say that they're providing a valuable service, even if they aren't communicating with the spirits, by helping those people come to grips with their loss, and gaining the comfort of some sort of closure. And still others claim that they're providing a service that's quite valuable, being an intermediary and speaker for the dead.
A somewhat interesting perspective on the subject of mediums comes from the transhumanist society, the people dedicated to proving that the human consciousness can be downloaded as a digital media type. If they're true, would talking to a downloaded personality be any different from use of a medium in the traditional sense? This may well become a case where life imitates art, and in the near future at that.