The problems of consciousness are many, and equally varied, but I think it would prove useful to put forth a rough description, nonetheless.
What it comes down to is the following: there is something that it is like to be a conscious being. You can describe the brain as much as you like and abstract away from the details, but all of these purely physical, third-person accounts seem to inevitably miss some crucial element, namely consciousness itself. Brains can instantiate causal relations, we know that. Brains can control behaviour, we know that, too. But how do physical brains give rise to subjective, mental contents? This is the most fundamental problem that all of the remaining issues derive from.
Everywhere else we look in the universe, we see physical entities and nothing more, we see objectivity. In a way, the entirety of scientific discovery, in crude form, is dependent on this, so how are we even to approach the problem at hand? All of our traditional tools of measurement, of explanation, of prediction rely on roughly deterministic, objective natures. Consciousness alone stands in the face of this, or at least it appears to. We would like to find a theory that accounts for this, but how is not at all clear. In fact, it could be said that conscious existence is one of the few remaining problems in science for which we do not even know how to ask the question. This should help to explain all of the contradictory accounts that are thrown around on a day to day basis. It helps explain why so many resorted, in desperation, to dualism. Life would be a lot simpler if there were some mysterious substance just out of reach that could do exactly what we need it to. But this is probably not the case, and most modern theorists understand this. Equally, it helps explain a great deal of the contemporary resistance to the downfall of materialism (or physicalism, if you prefer). Materialism is all that we know, it seems, are we not lost if we give up on it? From a more reflective point of view, though, and with time, I think these perspectives can be overcome.
To the cry of desperation of the materialist, think of this: physicalism is not all that you know—you know of your own conscious mind much more soundly, even if in an irritatingly limited manner. To the dualist, not all avenues of inquiry have been exhausted. In fact, we likely have barely scratched the surface of this enormously complex phenomenon. Where we go from here, I do not know, and it is probably anyone’s guess, but we at least have some idea of what we need to think about, and the work of others before us will allow us to avoid their pitfalls and misfortunes. This is, I think, the greatest question that lays before us as a species, and I think we may—finally—be prepared to tackle it head on.