Dualism is the doctrine that there exist two fundamentally different sorts of thing in this universe, which we might call 'mind' and 'matter'. It is usually discussed in the context of the mind/body problem - the puzzling fact that experiences, thoughts and feelings appear to us quite different from the physical things that by and large seem to make up everything else in the universe, even though there is clearly some kind of intimate connection between the mind and the brain.
Nowadays most scientists take a materialist* attitude to the problem, in the sense of holding that everything is matter. That is, the stuff of the universe consistently obeys the laws of physics, and anything we ever experience must ultimately arise from those laws, however many layers of explanation might be required in between. This is a form of monism, the idea that everything consists of variations on kind of stuff; the other main form of monism is idealism, which holds that there is nothing but mind, and all seeming physical manifestations arise from that.
Descartes thought he had some knock-down arguments for why thoughts and feelings must be fundamentally independent from the physical world, and he speculated that maybe our spirits interact with our bodies through the medium of the pineal gland. Descartes was surely a genius of world-shaking proportions, but all the same he thought a lot of silly things. In retrospect his arguments for some of them were kind of deranged.
Daniel Dennett is also plainly a genius, and I'm very much enjoying reading Consciousness Explained at last, but it happens that what he thinks is a knock-down argument against dualism is also pretty weak. His argument rests on the idea that if something interacts with physical matter, it is by definition a form of physical matter itself. Hence there is no way something could interact with our world without being subject to its laws of physics. The whole thing falls down if you consider the form of our interaction with computer games - we control avatars which may or may not obey the same laws as the rest of the game, and we ourselves are totally unaffected by in-game physics. Higher-dimensional or pure-thought beings interacting with our four-dimensional universe might well be in a rather similar position.
Could the world be something like a giant computer game, then? Our minds might exist in some other universe - perhaps they even lead other existences outside of the game, temporarily forgotten while we play through this life. Can we rule out this possibility? No - but I think we can come pretty close.
The biggest problem with any conception of the universe including a mind or self which doesn't arise chiefly from the actions of the brain is that our thinking is quite clearly modified on a basic level by things happening to our brains. If something else is doing the thinking, that other thing is clearly transformed somehow by things like brain injuries and drugs. Our 'players' - or 'spirits' to use the more traditional term - lose or gain certain faculties as a result of the bodies they're controlling. It's not inconceivable, but if the mind is neither the brain, nor something produced by the brain, then why would it be changed so profoundly by what happens to the brain?
It's a lot for a dualist conception of the cosmos to accommodate. It starts to look as though things have been specially set up so that it looks as much as possible like minds are actually offshoots of brains. It's quite conceivable that they could have been set up that way, of course - maybe our players have drugs injected into their brains when we take drugs, or maybe it happens to be a magical feature of the body-spirit link that our spirits lose the ability to handle language when certain parts of the brain get damaged (in which case what should we expect to happen when the whole brain is damaged beyond repair?). It seems highly likely that a more straightforward explanation could be found which supposes that the profound link between minds and brains is there because it has to be - because one is the result of the other.
All of this leaves us, of course, with the very interesting question of how our subjective experiences can possibly arise out of physics, chemistry and biology. It also leaves untouched the related question of how much of cognition is specifically down to the brain; it may be that our sense of self is an emergent feature of our embodiment, and it may be that cognition takes place on a significantly larger scale, or that thought-like processes occur in other types of physical system, as animism and pandeism always maintained. Serious exploration of these sorts of questions is only possible with some understanding of how consciousness arises in the first place.
Maybe some will still think that explaining conscious experience in terms of physical processes is such a tall order that there must be some other explanation entirely. However, the difficulty of imagining something is not always a strong argument for its impossibility, and it is not clear that the arguments against materialist explanations for the mind amount to anything more than that. This natural incredulity may well be dispelled by further study - and either way, explaining the known facts about interactions between mind and brain any other way is a pretty tall order too.
* It's probably worth flagging up, in case it's not obvious, that the philosophical meanings of 'materialism' and 'idealism' have almost nothing to do with what they mean in popular use.