The hard problem of consciousness is the problem of explaining why and how sentient organisms have qualia. how and why it is that some internal states are subjective, felt states, such as heat or cold, rather than objective states, as in the workings of a thermostat or a toaster (From Wikipedia)
...It is undeniable that some organisms are subjects of experience. But the question of how it is that these systems are subjects of experience is perplexing. Why is it that when our cognitive systems engage in visual and auditory information-processing, we have visual or auditory experience: the quality of deep blue, the sensation of middle C? How can we explain why there is something it is like to entertain a mental image, or to experience an emotion? It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. Why should physical processing give rise to a rich inner life at all? It seems objectively unreasonable that it should, and yet it does.
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The really hard problem of consciousness is the problem of experience. When we think and perceive there is a whir of information processing, but there is also a subjective aspect.- David Chalmers, Facing Up to the Problem of Consciousness (1995)