Qualia are the specific qualitative properties of our experience, as accessed by first-person introspection. The raw redness of the red you notice in your field of vision, for instance, as contrasted with the functional state of visually detecting light of wavelength 620–740 nm that an alien building a cognitive model of your behavior might initially construct. (Which is not to say that those two properties are non-identical. But if the two are identical, this must be discovered, not just stipulated.)

There are several popular arguments for the irreducibility of qualia (and, more generally, against the reliability of introspection as a method for directly reading off part of our world's ontology), which has made them controversial posits.

In a conversation about mindstuff, what Eliezer calls 'Reductionism' is what I'd call 'physicalism' — the doctrine that the mental (unlike physicsy stuff) is non-fundamental, that it can in principle be fully explained in non-mental terms. The 'reductionism' I think about (which I'll distinguish by making it minuscule) is the more specific doctrine that mental stuff — in this case, phenomenal properties, qualia — exists and is reducible. So a physicalist has to either eliminate or reduce every mental posit.

Eliminativists might insist that we reject qualia because the evidence for them is strictly first-person and introspective (phenomenological) rather than sensory and in-the-world (scientific). Or an eliminativist might think that we have strong prior grounds for accepting physicalism, but that reductionism is a doomed project, say, because of the Mary's Room argument.