Dualism is the position that mind and body are in some categorical way separate from each other, and that mental phenomena are, in some respects, non-physical in nature. It can be traced back to Plato, Aristotle, and the Sankhya and Yoga schools of Hindu philosophy, but it was most precisely formulated by René Descartes in the 17th Century. Descartes was the first to clearly identify the mind with consciousness and self-awareness, and to distinguish this from the brain, which was the physical seat of intelligence.
Dualism appeals to the common-sense intuition of the vast majority of non-philosophically-trained people, and the mental and the physical do seem to most people to have quite different, and perhaps irreconcilable, properties. Mental events have a certain subjective quality to them (known as qualia or "the ways things seem to us"), whereas physical events do not.
There are three main Dualist schools of thought:
Substance Dualism (or Cartesian Dualism) argues that the mind is an independently existing substance - the mental does not have extension in space, and the material cannot think. This is the type of Dualism most famously defended by Descartes, and it is compatible with most theologies which claim that immortal souls occupy an independent "realm" of existence distinct from that of the physical world.
There are three main types of Substance Dualism:
Interactionism, which allows that mental causes (such as beliefs and desires) can produce material effects, and vice-versa. Descartes believed that this interaction physically occurred in the pineal gland.
Occasionalism, asserts that a material basis of interaction between the material and immaterial is impossible, and that the interactions were really caused by the intervention of God on each individual occasion. Nicholas Malebranche was the major proponent of this view.
Parallelism (or Psychophysical Parallelism), holds that mental causes only have mental effects, and physical causes only have physical effects, but that God has created a pre-established harmony so that it seems as if physical and mental events (which are really monads, completely independent of each other) cause, and are caused by, one another. This unusual view was most prominently advocated by Gottfried Leibniz.
Property Dualism maintains that the mind is a group of independent properties that emerge from the brain, but that it is not a distinct substance. Thus, when matter is organized in the appropriate way (i.e. in the way that living human bodies are organized), mental properties emerge.
Epiphenomenalism, which asserts that mental events are causally inert (i.e. have no physical consequences). Physical events can cause other physical events, and physical events can cause mental events, but mental events cannot cause anything, since they are just causally inert by-products of physical events which occur in the brain (i.e. epiphenomena) of the physical world. This doctrine was first formulated by Thomas Henry Huxley in the 19th Century, although based on Thomas Hobbes' much earlier Materialism theories.
Predicate Dualism argues that more than one predicate (how we describe the subject of a proposition) is required to make sense of the world, and that the psychological experiences we go through cannot be redescribed in terms of (or reduced to) physical predicates of natural languages.