Functionalism emerged in the 1960s from cognitive science, and replaced identity theory and central state materialism as the dominant philosophy of mind. It was developed mostly within the materialist tradition, and is therefore generally considered to be anti-dualist, though this is not necessarily a feature of functionalism.
Functionalism first emerged as a criticism of type-type identity
theory, claiming that it was wrong to locate mental phenomena in matter
because that implies:
Functionalism's central thesis is that mental phenomena depend not upon the matter in which they 'reside' but in their organisation; i.e. mental states are functional states. A functional analysis of mental phenomena would therefore explain the working of the component parts, or functions, of a psychological organism, and their overall organisation. The mind could therefore be explained in terms of (admittedly complex) flow charts. The mind, and mental faculties, are defined in terms of what they do, rather than what they are, just as one would define 'crank shaft' as 'something that opens and closes valves', rather than identifying the definition with a particular physical object.
Functionalism solves one of the most damaging problems of behaviourism, which is that one could have two stimuli cases providing quite different responses, or indeed several cases of the same stimuli-response relationships with different associated mental states. With functionalism, having the same inputs and outputs needn't mean having exactly the same mental phenomena, since the functions that process the inputs and provide the selfsame outputs can differ. This implies quite a startling conclusion: if one could map out the entire functional process of one thought, one could make a computer 'think' in an identical fashion, and so it is quite conceivable that a machine could love a human, and vice versa.
On this theme, functionalism can be summed up as follows: mind is to brain as software is to hardware. Two systems can be computationally equivalent by physically different.
Is functionalism dualist, materialist or idealist?
By defining itself in terms of function rather than in terms of its substantial instantiations, functionalism remains ontologically liberal (identity theory was thought to be 'chauvinistic') and so avoids the problems of what kinds of beings can have mental phenomena, and of dualism. Functionalism could be considered compatable with a very liberal token-token identity theory, however, as one could claim that each mental state is identical to a particular physical state (e.g. my pain Y, happening right now, is identical to brain state X) but that mental states in general needn't be thought of as identical to physical states in general (e.g. pain Y is always identical to brain state X). An identity theory this liberal, however, without the additional ideas of functionalism, becomes more or less meaningless as nothing that it can tell us can ever be proven in any sense (in positing an identity one needs to be able to know of each honomonous entity distinctly, and show a pattern of coincidence with reasons why it is more than coincidence; here one can find no pattern).
It is this distinction between functional states and physical states that leaves functionalism open to forms of dualism, in particular property dualism. Mental phenomena can be thought of as functional properties of brains, with sufficient detachment for entirely different functional states to be associated with the same physical state, and vice versa. In fact, it is conceivable that functionalism might fit with idealist theories. There are no contradictions in functionalism in supposing that a conscious being might have a mind but no brain, and so it is logically possible. Materialism cannot therefore be considered a conceptual truth based on functionalism alone.
Functionalism and consciousness
According to functionalism, one might be tempted to say that a conscious system is a computational system that is capable of posessing functional properties. If we consider Bill Gates, aside from his obvious physical properties, the kinds that any physical thing might have (e.g. his weight, height, his particular chemical composition), one can attribute various properties to him: he is Chairman of Microsoft; he is a computer programmer, etc. Each of these properties are functional properties, and by virtue of having them he is conscious.
Of course, going by that definition, a pocket calculator could be said to be conscious. So it seems that to answer the problem of consciousness, functionalism needs more than a theory of functional processes and properties. Intuitively, consciousness seems linked to the ability to posses functional properties that are capable of producing qualitatively different mental phenomena, what some philosophers refer to as qualia.
But it seems entirely possible, and coherent with functionalism, to suggest that two people could experience entirely different qualia with exactly the same equipment, e.g. two people with the same visual "equipment" could see a block of colour; one could see "red", the other "green" (putting linguistic considerations aside). This phenomenon is known as inverted qualia, and poses no problems for functionalism as such. In fact, in Inverted Earth Ned Block pointed out that one could conceivably render someone unconscious, invert their visual equipment and put them into a room that is identical to that of a second person except that all colours are reversed; in this situation, their qualia would be identical, since what it would be like to experience the colours would be identical for both people, despite their hardware and the reality they are seeing being opposite to one another.
However Block also raised several problems with qualia that affect functionalism in particular. The first is to say that if we accept that inverted qualia pose no problems to functionalism as such, then we must also accept the logical possibility that my visual qualia in a given situation might be identical with your auditory qualia in a given situation, since there is no necessary connection between qualia, the person's hardware and the environment that is 'causing' the experiences. This is both perplexing and attractive to those who feel materialism to be too inhuman, since it suggests that my listening to Beethoven's 5th symphony and your reading a novel set in the Napoleonic wars could evoke exactly the same mental phenomena; what is is like for me to listen to the music could be identical to what it is like for you to read the book. Romantic as this seems, it does make explaining functional states and properties extremely difficult, as you can no longer use the hardware nor the environment as a reliable reference point.
There may also be mental states that aren't public, such as unconscious states, and so that functionalists cannot accurately describe nor explain, at least not objectively, since they cannot get any reliable information by interacting with the owner of the states, they can't look at the brain states, and they can't rely on the subject's environment.
One can also posit the existence of zombies, unconscious beings that have the same behaviour and the same brain states as a conscious being, but no qualia. The functionalist ought to be able to distinguish a zombie from a conscious person, but given that all of the public information seems identical to that of a conscious person, how could one ever know? How could we know everyone else isn't a zombie, or that we ourselves aren't zombies? By being so liberal about qualia and their relation to the brain and the subject's environment, functionalism solves a lot of the problems associated with behaviourism and identity theory, but it also opens the door to extreme forms of scepticism, and makes it extremely unlikely that it could deliver any scientific explanations of the functioning of a specific mind-brain.