What is it that we're aware of when we undergo visual experiences? It's natural to say that we're aware of the world. The red apple I see is on my desk, not in my head. But experiences themselves are surely in the head. The issue may be clarified by conceiving of experience in terms of representation. A book can be about things beyond the confines of its pages, and similarly for the conscious representations in our minds.
One of my experiences is a representation of an apple. I perceive the apple as being red: this is how it is represented in my experience. My mind represents the external world. I am introspectively aware of the content of the experience, i.e. the external world, and not the vehicle or representation itself. To attend to the "redness" in my visual experience, I direct my attention outward, to the apple. I cannot see into my own mind, even though that's where the event of my seeing occurs.
Our talk of conscious experience risks conflating two distinct objects: the qualities (properties) of the vehicular experience or representation itself, versus the qualities represented in the experience. It's plausible to think that we can access only the latter. For information to be available to us, it must be represented in our minds. There are facts about our minds that are not so represented, and so remain inaccessible to us. Thus cognitive psychologists surprise us by revealing how our minds really work.
This includes experience itself: there may be facts about it of which we remain unaware. This is plausibly true of the temporal properties of our experiences, for instance. It seems to me that I experience A before experiencing B, but in actual fact all I have access to is my mental representation as of "A followed by B". The representation need not be accurate: I might in fact have an experience [of B], followed by a later experience [of A], without being aware of this objective sequence at all. If the information isn't itself represented, then I cannot access it. Merely being true does not suffice. I don't really have a "mental eye" in my head to tell me what's going on in there, independently of what eventually enters into a conscious representation. Our introspective capabilities are hence severely limited.
So what are qualia (or the "qualitative character/properties" of experience)? The term is ambiguous. Some use it to denote those properties of which we are aware. When I feel pain in my foot, what I am aware of - what is represented in experience - is a property of my foot, say tissue damage. On this odd way of speaking, I can have a pain without feeling it, and I can feel painful hallucinations when there is no real pain (tissue damage).
The more relevant use of the term concerns the properties of our experiences ourselves (i.e. the vehicles, not the contents). Painful experiences have a particular phenomenal "feel" to them, and this is unaffected by whether or not they are veridical (in the sense of accurately reflecting one's bodily condition).
Curiously, we do seem to have infallible introspective access to this phenomenal property, of "what it is like" to have the experiences we do. My earlier remarks might be taken to imply that we must thereby represent this further fact of what it is like. But that would leave our phenomenal infallibility a mystery, since misrepresentations are always possible. Instead, I think, our access to the phenomenal feel must derive from the original contents themselves. The fact that I consciously represent [pain in my foot] entails that I have access to the represented contents. I can know that I represent [pain in my foot], because this content - [pain in my foot] - is consciously accessible to me. It is the content of which I am directly aware. And while the content itself might misrepresent (there might not really be anything wrong with my foot), there can be no mistaking my conscious access to the content, and it is the latter fact that constitutes phenomenal feel.
But enough of my untutored reflections on the philosophy of mind. It's quite a confusing area! Hopefully others will clarify matters further in the comments...