Qualia are what experiences are like from the inside. The singular is quale. Seeing a ripe tomato involves a visual state with a particular character, and that character is what the word names.

Why they are a problem

Every other property science explains gets analyzed into structure and function: what a thing is made of, how it is arranged, what it does. Qualia appear to resist that, because a complete functional account of color vision seems to leave the question of why it looks like anything untouched. This is the core of the hard problem.

The two arguments

The knowledge argument. Frank Jackson's 1982 thought experiment imagines a scientist who knows every physical fact about color vision but has only ever seen black and white. On leaving the room and seeing red for the first time, does she learn something new? If she does, there was a fact she did not have, and physical facts are not all the facts.

The inverted spectrum. Suppose your experience of red is what mine is of green, consistently, throughout. Every behavioral and functional test would come out identical, because we both stop at the same lights and use the same words. If that is coherent, qualia are not fixed by function.

The replies

Physicalists respond in several ways. The scientist gains an ability rather than a fact — knowing how to recognize red rather than knowing a new truth. Or she learns an old fact in a new way, as someone can know that water is present without knowing it as H2O. Daniel Dennett argues the concept is confused and dissolves under examination, and that the intuitions supporting it come from the thought experiments rather than from anything observed.

Why it stays open

Nobody disputes that experiences have a character. The dispute is whether that character is something over and above the physical and functional story, and both sides agree no experiment settles it — which is itself the interesting fact about the problem. See mind-body problem and materialism.