The philosophy of mind runs on thought experiments the way a car runs on fuel, sometimes brilliantly, sometimes producing a lot of smoke. Here are the three that have most defined the contemporary debate.
Qualia: this is just the term for the subjective, felt qualities of experience, the redness of red, the painfulness of pain, the taste of coffee. The claim is that qualia are real features of experience that resist any purely functional or physical description. You can describe the functional role of pain (it causes withdrawal, signals damage, interacts with attention) without capturing what pain actually feels like. The felt quality is something over and above the functional role. Physicalists dispute this, they say qualia are not over and above the physical story, and the intuition that they are is a cognitive illusion. This debate has been running hot for decades.
The Mary argument (Frank Jackson, 1982): Mary is a brilliant neuroscientist who has been raised in a black-and-white room. She knows every physical fact about color vision, the wavelengths, the neural pathways, the functional roles. She has never seen red. Then she is released from the room and sees a red rose for the first time. Does she learn something new?
Most people's intuition is: yes, obviously she learns something new, what red looks like. And if she learns something new when she leaves the room despite already knowing all the physical facts, then there are facts about experience (what red looks like) that are not physical facts. Therefore physicalism is false.
The materialist responses are many and clever: perhaps Mary doesn't learn a new fact, she just gains a new ability (the ability to recognize red). Perhaps "all physical facts" in the thought experiment is incoherent, a complete physical description would already include whatever is needed for full color knowledge. Perhaps the new knowledge is just a different mode of representation of the same fact. None of these responses has been universally accepted.
Philosophical zombies (Chalmers): imagine a being physically identical to you in every way, atom for atom, neural firing for neural firing, but with no inner experience. No qualia. No one home. A zombie walks, talks, processes information, and reports having experiences, but there is nothing it is like to be it. Is such a being conceivable?
Most people's intuition is: yes, it seems conceivable. And Chalmers argues that conceivability is evidence for possibility: if zombies are conceivable, they are possible; and if they are possible, then consciousness is not logically entailed by physical structure. Therefore consciousness is something over and above the physical, not reducible to it.
Physicalists resist at the conceivability-to-possibility move. Just because you can conceive of something doesn't mean it's metaphysically possible, you can conceive of water that isn't H₂O, but water is necessarily H₂O. Perhaps consciousness is necessarily identical to certain physical processes, and our intuition that zombies are conceivable just reflects our ignorance of this necessity, not a genuine gap in reality.