"Dualism" and its modern nuances
To my understanding, most people are dualists of some sort since: (1) we have a powerful intuition that mental events are real in some sense and not just an artifact of physical processes, and (2) the properties of mental phenomena are extraordinarily difficult if not impossible to describe in purely physical terms. At the very least, most of us acknowledge that the very fact that we are so compelled to talk about minds the way we do means that they have their own kind of description, and thus their own properties in some sense. There are an awful lot of nuances possible within this grand view, however.
To clarify a point about “dualism,” consider some conventional alternatives for comparison, and focus on good examples (definitional entries copied from http://www.artsci.wustl.edu/~philos/MindDict/ )
Dualism in general, with classical examples: http://www.artsci.wustl.edu/~philos/MindDict/dualism.html
Most generally, the view that reality consists of two disparate parts. In philosophy of mind, the belief that the mental and physical are deeply different in kind: thus the mental is at least not identical with the physical.
That is a very different proposition from the claim that that they are “independent” or saying that “the mental cannot be reduced to the physical.” Not to mention that “independent” and “reduce” can mean vastly different things in different contexts.
First important question: are mental and physical “different objects” or are they “different properties of the same object?” Answering that gets you off the back of people who think you are making “Descartes’s error.”
Second question (nuance): If mental and physical are different properties of the same object (presumably brain, or body and their associated processes in the materialist tradition) then what is the relationship of those properties. Is it causal, and in what way? Is it descriptive or explanatory (does it provide a way of describing things that cannot be achieved otherwise) ?
Third question (nuance): Can one be subsumed entirely within the other (“supervenience”)?
Fourth question (nuance): Can one be explained entirely in terms of the models we use to describe the other, either in principle or in practice? What sort of translation is possible and how?
The view that the mental and the physical comprise two different classes of property that are coinstantiated in the same objects
According to property dualism, even though mental properties are totally different than physical properties, they are nonetheless all properties of the same kinds of objects. Thus, for example, a single object instantiates the property of my being six feet tall and my believing that the Eiffel tower is in France. Property dualism is compatible with the token identity thesis, but not the type identity thesis. Property dualists are typically, if not unanimously, anti-reductionists about the mental, which is to say, they deny that it is in-principle possible to translate mental predicates into physical predicates.
The view that the mental and the physical comprise two different classes of objects: minds and bodies.
Perhaps the most famous proponent of substance dualism was Descartes, who cashed out the distinction between minds and bodies as follows: minds are things that think but lack spatial magnitude, and bodies are things that have spatial magnitudes, but don't think. Different substance dualists may disagree as to how best to define what's essential to being mental and physical, but they do agree that the difference in question is one of objects, not properties. So, for example, my belief that the Eiffel tower is in France and my being six feet tall are properties of different objects, i.e., my mind and my body, respectively.
If you see mental and physical as having some causal connection, it can easily be distinguished from the various forms of parallelism, the classical forms of dualism which do not connect them directly in any causal way (I suspect they have variants that still have modern adherents):
Occasionalism: Espoused by Clauberg, de la Forge and Malebranche, occasionalism entails the contention that everything is devoid of causal efficacy and that God is the only truly causal agent. So, for example, placing your hand on a hot stove is does not cause pain, but is rather an occasion for God to cause the mental state of pain. So, not only mind/body interactions, but all causal interactions become the work of God.
Preestablished harmony: This doctrine was formulated by Leibniz and is basically Cartesian interactionist dualism without the interaction. Thus, rather than causal interaction, God has provided setup in which the mental and physical are synchronized so as to provide this appearance. However, it should be noted that Leibniz himself was not a dualist: for him there were no physical substances, these were just appearances. Nevertheless, this position is often considered a possible dualistic solution to the mind-body problem.
Both occasionalism and preestablished harmony are considered instances of parallelism: the view that the mental and physical realms co-occur but are not causally connected.
More usefully, if you do see M and P as having some causal connection, then it is very helpful to distinguish it.
supervenience - A set of properties or facts M supervenes on a set of properties or facts P if and only if there can be no changes or differences in M without there being changes or differences in P.
In other words, the mental supervenes on the physical if it is impossible to have mental events that have no physical correlates whatsoever. This is one of the most useful and nuanced ways to talk about “independence.” The term was introduced by the late Donald Davidson, but Jaegwon Kim has written some of the most interesting modern analysis about supervenience in mind-body questions.