Pronounced either kwa-lee or kwa-la. The plural is qualia, pronounced kwa-li-a (that first 'a' can be either long or short, depending on personal preference).
A quale is a mental state or mental event with a distinctive subjective character.
These are all those aspects of thoughts that you can 'see' or 'feel' in your head. ("The introspectively accessible, phenomenal aspects of our minds".)
That is the simple definition. Philosophers spend a lot of time arguing what exactly qualia actually are, and what they can be applied to, and if they actually exist. The status of qualia is important in philosophy of the mind and consciousness. (See The mind-body problem).
There's no way I can do justice to all qualia-related issues, but I'm going to prattle on a bit so that you can get a good handle on what exactly (Tem42 thinks) a qualia is, and how they are used in philosophy.
First up, here's an example of a philosophical issue in which it is useful talk about qualia. It is called the inverted spectrum:
It is possible that when you and I look at a color, we actually percieve different colors. But as long as we consistently label colors, we would not know that one word refers to two different colors. For example, imagine that we were both to look at the same apple; you see the apple as red. But when I see the apple, I see what you would call 'blue'. Since I have always heard everyone call this color 'red', I also call the apple red. Every time I see something blue (to me), I call it by the word 'red', because everyone else does so.
When talking about the inverted spectrum, we are (probably) not questioning whether the apple is reflecting different light waves for me than it is for you. We are questioning whether these light waves are causing the same mental experiences in both of us, that is, if the qualia caused by 'red' light is the same for both you and I.
When talking about such things it is useful, if not necessary, to use some word or term that is equivalent in meaning to qualia.
But wait! Some people will deny that qualia exist! How can this be - there most certainly is some mental experience going on in my mind - how could they deny this? Well, leaving minor (and not so minor) differences between definitions of qualia beside, there are at least two ways that the definition given above could be said to describe something that doesn't exist.
First off, some people will read the definition above and claim that they have never experienced any such thing. Maybe they honestly do not experience this qualia, maybe they are interpreting the definition differently than I am, maybe I'm schizophrenic and seeing things others don't.
Secondly, maybe they're not disputing the definition of 'qualia'. Maybe they're just using a different definition of 'exist'. The everyday definition of 'exist' is probably something along the lines of "look, an x. I guess x's exist". While I am certain that qualia must exist in this sense, there is another sense of exist.
When building an ontology, a person tries to enumerate everything that exists and the forces acting upon them. Since an ontology, if it is complete, should include everything that exists, anything that doesn't appear in this ontology can be said to not exist. This leads to all kinds of weirdness -- for example, if I hold that the complete set of basic building blocks of the universe consists of subatomic particles, I can say that (in my ontology) tables don't exist. That would not mean to imply that we don't see tables in our everyday lives, but it does mean that once I've listed every bit and piece of the universe, I wont have a table listed among those pieces.
Likewise, I could build an ontology without any qualia. Moreover, I can then use my ontology to explain how things like qualia and tables come about. When I do this, I don't ever need to use the word qualia; I might instead use words like 'chemicals' and 'synapses' and 'neurons'. And if I did this, I might decide that having people use the word qualia is only going to complicate their understanding of this universe. To put it another way, mental events might be best understood as completely physical events, not mental, and any mumbo-jumbo suggesting otherwise should be thrown out.
Personally, I am more certain of my qualia than I could possibly be of anything else. I might, for example be a brain in a vat being fed images by a mad scientist (think The Matrix). In this case, this computer that I'm typing at does not exist in any real and objective sense. What does exist is my experience of the computer; the feel of the keys under my fingertips, the glow of the screen, etc. These are the qualia, and they are most certainly here. The existence of qualia is the one thing in the universe that I can be 100% sure of, because they are the only thing that I experience directly.