The core idea: mind as function, not matter
Functionalism says that what makes a state a belief, a desire, or a pain is the causal role it plays: the pattern of inputs that cause it, its interactions with other mental states, and the behavior it tends to produce. Pain, for example, is whatever state is typically caused by bodily damage, causes distress and the belief that something is wrong, and leads to wincing and avoidance. By this logic a mental state is like a mousetrap or a carburetor, defined by its job rather than its material.
This leads to the thesis of multiple realizability: the same mental state could in principle be realized in neurons, in silicon, or in some entirely different substance. What matters is the organization, not the stuff.
Who developed it
Functionalism grew up in the 1960s and 1970s as an answer to two earlier views, behaviorism and the mind-brain identity theory. Hilary Putnam introduced machine state functionalism, modeling the mind on a Turing machine whose states are defined by rules mapping inputs and current states onto outputs and new states. Jerry Fodor developed the version tied to cognitive science, and David Lewis defended an analytic version on which mental terms are defined by their place in a network of commonsense causal truths.
Because it treats the mind as substrate independent, functionalism became the philosophical backdrop for artificial intelligence and cognitive science. If having a mind is a matter of function, then a machine that duplicates the right functions should, in principle, duplicate the mind. That implication is exactly what makes functionalism central to debates about whether AI could be conscious.
The qualia objections
The best-known challenge concerns conscious feel, or qualia. Critics argue you could reproduce a person's full functional organization while leaving out experience entirely. Ned Block's Chinese nation thought experiment imagines the whole population of China wired up to mimic the causal roles of someone's neurons; the system seems to satisfy the functional description, yet it is hard to believe the nation as a whole feels anything, an absent qualia worry. A companion objection, inverted qualia, imagines two people who are functionally identical but whose color experiences are systematically swapped.
Frank Jackson's knowledge argument and John Searle's Chinese room press related points against pure function. Functionalists reply in various ways, from denying that the scenarios are genuinely possible to arguing that our intuitions about them are unreliable. The exchange remains one of the liveliest in philosophy of mind.