What qualia are
A quale (the singular of qualia) is the introspectible, subjective character of an experience: not the wavelength of light hitting your retina, but the particular way red looks to you; not the tissue damage in your hand, but how the pain feels. Philosophers use the word for the felt textures of perception, bodily sensation, mood, and emotion. When people say there is something it is like to taste a lemon or to hear a cello, qualia are the something.
The concept is meant to isolate the part of the mind that resists neat functional or physical description. You can specify, in full, what a color-detecting system does: which inputs trigger which outputs and behaviors. Yet that specification seems to skip the presented quality, the vivid look of the color itself. Qualia are often characterized as the properties that would remain unexplained even after every function had been mapped. Frank Jackson once described them as features of experience that no amount of purely physical information includes.
Where the term comes from
The word derives from the Latin qualis, meaning of what kind. C. S. Peirce used quale in a related technical sense as early as 1866, but the modern usage traces to the American philosopher C. I. Lewis, who in his 1929 book Mind and the World Order tied qualia to the given elements of sensory experience. The term gathered force in the second half of the twentieth century as debates about the mind and the brain sharpened.
Thomas Nagel's 1974 essay What Is It Like to Be a Bat? became the touchstone. Nagel argued that even complete knowledge of a bat's echolocating brain would not tell us what the bat's experience is like from the inside, because subjective character is tied to a particular point of view that objective science does not capture. The essay did not use the word qualia heavily, but it framed the puzzle that the term now names.
Why qualia are contested
Qualia sit at the center of the mind-body problem. If experiences really do have intrinsic felt qualities that physics leaves out, then a purely physical account of the world would be incomplete, which is why qualia feature in arguments against physicalism such as Mary's Room and the conceivability of philosophical zombies.
Not everyone accepts the premise. Daniel Dennett, in his 1988 paper Quining Qualia, argues that the ordinary notion of qualia bundles together several properties (ineffable, intrinsic, private, immediately apprehensible) that nothing in the world actually satisfies, so the concept should be retired rather than explained. Defenders reply that Dennett has explained away the very datum a theory of mind is supposed to account for. The live question is whether qualia are a genuine feature of reality that science must someday accommodate, or an artifact of how introspection describes itself.