The Multiple Realizability Objection is an argument against
type materialism (which you may want to read before continuing), a
materialist view in
philosophy of Mind. It works like this:
Imagine that whenever a human feels pain, he has c-fibers firing. In such a case, every mental state of type 'pain' is identical with some physical state of type 'c-fibers firing'. Thus, anything that can feel pain must necessarily have c-fibers. Now, imagine our friend Marty. Marty happens to be a Martian. Having evolved through an entirely separate evolutionary chain from our own, Marty has no c-fibers. Therefore, if type materialism if true, Marty can not feel pain. But of course, Marty can feel pain. Therefore, type materialism is false.
This is currently the most convincing argument against type materialism. The reason it is called the 'multiple realizability' objection is that it centers on the fact that mental states are multiply realizable: in humans, the mental state 'pain' may be realized via the physical state 'c-fibers firing', whereas in Martians, it may be realized via the physical state 'm-fibers firing'. One reply to it is to say that Marty does not, in fact, feel pain as we know it; in fact, he feels m-pain, which may or may not be caused by and have similar symptoms to human pain. And m-pain is identical with some type of physical property unique to Martians. Token materialism is not vulnerable to this objection.
See also: Philosophy of Mind, type materialism.
All material taken from whatever I managed to absorb from my Philosophy of Mind class at the University of Rochester during Fall 2003.