General idea: the essential nature of a mental state such as pain (i.e. the characteristics that all pains share by virtue of which they are pains) is to occupy a certain functional role, in particular a causal role.
More informally: mental states are defined by the "causal work" they do.
What's a functional state? Usually this is taken to be a state defined by its relations to inputs, outputs, and other (also functionally defined) states.
This sounds circular, because every functional state is defined in terms of other functional states. However, we can define all of them at once.
Versions of functionalism:
machine functionalism: This is sometimes put as the thesis that minds are Turing machines. Kim himself puts it this way. But (I claim) this is very misleading. The key idea of machine functionalism is that thinking is computation, that mind is to brain as software is to hardware, so that thinking and other mental processes are the operation of a program that "runs on" the brain. However, this doesn't require thinking that the architecture of the human mind is the same as that of the Turing machine! The Turing machine is one model of computation, but not the only one; it's equivalent to other models of computation in the sense that the same functions are computable by Turing machines as by other computational models, but it's not equivalent to other models in the stronger sense that the computations work the same way. Machine functionalism says that thinking is computation, but it need not be committed to the idea that it is Turing machine computation.
Why should we call this "machine functionalism"? Because the key idea is that humans are computing machines of some sort. Just as there are typewriters that all fall under the classification "typewriter," and all share states like "being in upper-case mode," despite having radically different physical constructions, so there are computing machines that all fall under the general classification "computing maching" or "computer," and all share states like "inferring that Q from P and if P, then Q," despite radically different physical constructions. In particular, human beings, like Macs and PCs, are computing machines.
causal-theoretical functionalism: mental states are defined in terms of a psychological theory.
which psychological theory? Two versions:
(a) common-sense psychological theory. (This can be regarded as defining common-sense psychological terms like belief, desire, etc.)
(b) scientific psychological theory.
[a different, orthogonal distinction: role functionalism vs. realizer functionalism. This is a fairly subtle and technical distinction. In a nutshell, role functionalism says that mental state = functional role while realizer functionalism says that mental state = realizer of functional role. Let's cash this out for a particular example. For the sake of the argument, let's say that the functional role associated with pain is: being a response to harm, leading to avoidance behavior, tending to cause people to dislike things that cause it, etc. Call this R (for "role"). For both kinds of functionalist, x is in pain iff x has some property or other that has role R. But what is the pain itself? For the role functionalist, my pain is my having some specific property or other that has R. For the realizer functionalist, my pain is the specific property that has R in me (and other creatures like me). So realizer functionalism, unlike role functionalism, is compatible with "domain-specific reductions": we can say that pain in humans is X, pain in Martians is Y, pain in robots is Z, etc.]
Over behaviorism: according to functionalism, psychological states are internal states that cause behavior. For (logical and metaphysical) behaviorism, by contrast, mental states are behaviors, not internal causes of behavior. (So in particular, functionalism can allow for the possibility that a surgical patient given curare and an amnestic drug feels pain even though they never exhibit any behavior associated with that pain.)
Over the identity theory: functionalism allows for the multiple realizability of mental states. The same functional state can be realized by various physical states.
The beginning point for discussion of the identity theory is the existence of systematic and extensive relationships between mental phenomena and neural phenomena.
There is a lot we don't know about these relationships, but there are also many things we do know.
Problems for Functionalism
Background point: recall the distinction between intentional and phenomenal states.
Intentional mental states are representational.
Phenomenal mental states are characterized by feelings, or "what it's like," or qualia.
Some states are intentional but not necessarily phenomenal: "dispositional" belief, desire
Some states are phenomenal but not intentional: "raw feels," perhaps itchiness or sadness
Some states are both: "occurrent" belief, sense perception
1. Problems of absent and inverted qualia. (Kim 179-181)
absent qualia: it seems theoretically possible to build something that has the same functional architecture as you but which has no qualia. If so, then a functional account of qualia can't be correct. (Philosophers use the term "zombie" to mean a being which is functionally like you or me, but has no qualia.)
inverted qualia: it seems theoretically possible to build a functional duplicate of you, but which has different qualia than you do. If so, then, again, a functional account of qualia can't be correct. The usual example here is the possibility of an inverted spectrum. [It's at least plausible that there may be real-life examples of this: Martine Nida-Rümelin has a lovely example to illustrate this.] Kim has a similar example in which pain and itching are switched (his Cross-Wired Brain example).
Possible responses: (a) incoherent; there can't really be functional duplicates with different qualia; (b) perhaps functionalism is right about intentional phenomena, while the old-fangled identity theory is right about qualia.
(Does this show -- if correct -- that a functional account of pain can't be correct? Depends on whether 'pain' refers to qualia, the feeling of being in pain, or rather to something that gives rise to qualia.)
2. Chinese Room argument (Kim 160-165)
This is John Searle's argument designed to show that functionalism can't account for intentionality. You could think of it as an "absent intentionality" argument.
3. Problem of inflexibility.
Kim points out that causal-theoretical functionalism seems to face the problem that if the theory is false, then no terms for mental states refer to anything. [This doesn't seem all that serious to me, and David Lewis anticipated it in the articles Kim cites. His suggestion: terms for mental states refer to the real-world state that comes the closest to satisfying the theory.]
4. Problem of holism.
The problem with defining all mental states at once is that the definition of each depends on the definitions of all the others. (see Kim, p 138)