Descartes set the problem in its modern form, and he set it as a genuine difficulty rather than a solved question. His argument for the distinction was that he could doubt he had a body but could not doubt that he was thinking — so mind and body must be different in kind. Mind is the thing that thinks and has no spatial extension; body is extended in space and does not think.
Then comes the difficulty he could never resolve, pressed on him sharply by Princess Elisabeth of Bohemia in their correspondence: how does an unextended, immaterial mind move a physical body? Physical causation as we understand it involves things with mass, position, and force. A thing with none of these has no evident way to push anything. Descartes gestured at the pineal gland as the site of interaction, which relocates the question without answering it — whatever happens at that gland, the same puzzle reappears at a smaller scale.
Physicalism cuts the knot by denying the split. If mental states simply are brain states, interaction is no more mysterious than one brain process affecting another. This position has the weight of modern science behind it: every advance in neuroscience tightens the correlation between mental and neural events, and damage to the brain reliably damages the mind in ways that would be surprising if the mind were independent of it.
But physicalism inherits a different difficulty, and it has proved remarkably durable. Even granting a complete neural account, there seems to remain a question it does not answer: why is any of that processing accompanied by experience? A system could, apparently, process light wavelengths and produce the word "red" with no felt redness anywhere. That intuition drives the hard problem of consciousness, and it is what thought experiments like Mary's Room and philosophical zombies are built to test.
Functionalism attempts a route between. What makes something a mental state, it argues, is the role it plays — what causes it and what it causes — not the material realising it. Pain is whatever is caused by damage and causes avoidance. This has elegant consequences, including that minds could in principle run on non-biological hardware. Critics respond that the theory captures everything about mind except the part that needed explaining: a system could occupy every functional role perfectly with nothing whatever going on inside. Property dualism and panpsychism take that objection seriously in different ways. The honest summary, four centuries after Descartes, is that the problem remains open.