There are various streams of Christian teachings on this subject, many of them stemming back to the very beginnings of Christianity. The main groups of teachings with some variations can be summarized and labeled as follows:

One Part Views
Monist, Psudeo-Dualist

From Mono = One - the understanding that human beings are comprised of just one part, a body. It is a materialistic view and either rejects or ignores the idea of any immaterial part of the human being. Often held as reactionary against some of the perceived divisionist teachings of the 2 and 3 part views.

This view emphasizes the holistic nature of human beings to the point of seeing them as comprised of a single part, namely a body. It is often held incompletely though when it comes to issues of the intermediate state. This view doesn't see that an intermediate state requires that there is more than just a body to the human being. It also doesn't acknowledge that the soul life can be anything more than is contained in the brain which will die and decompose.

A.A. Hoekema in his book "Created in God's Image" is an example of the monist or pseudo-dualist position. To the best of my knowledge this kind of view has only recently arrived on the scene, it is only materialism with christian trimmings. Also in this category are pseudo-dualist views that say that mind and body are just two ways of looking at the same thing ie the brain.

Two Part Views
Dualist, Dichotomist, Bipartite

The understanding that the human being has a body but also an immaterial constituent part or a "Soul", "Spirit" or whatever you want to call it. Dualism expresses the important idea that the individual continues to exist beyond death. It also usually affirms the idea of an intermediate state between death and the resurrection of the human body. It does recognize the requirement of there to be something beyond just the brain that carries human identity and personality between the demise of the body and the resurrection.

In the past some Bipartite views have been divisionist, teaching the philosophical idea that the spirit is somehow "higher" than matter and thereby leading people to devalue their bodies. This view has also lead to an undue emphasis that as Christians we must be exclusively concerned with the souls of people. However, this need not be so.

The Bible affirms both the value and importance of the human body. The Bible also affirms the importance of the whole of the human being, not just any particular part. More Biblical bipartite views affirm both these important issues and see man as an integrated whole comprised of two distinct parts that come apart when the body dies.

Dualists do not however recognize any distinction between 2 kinds of immaterial things within the human being ie a soul and a spirit. Important to this issue is how believers see spirituality and their relationship with God.

Dualists recognize that scripture requires a part of us that carries human personality and identity between the death and resurrection of the body. They thereby arrive at an understanding of what they call the soul. They are sometimes confused as to what relationship that means human beings are meant to have with God during this life. This means that people with Dualist philosophies sometimes have confused conceptions of spirituality.

Three Part Views
Trialist, Tripartite

From Tri = three partite = parts. The understanding that the human being is comprised of three structurally and functionally different but integrated parts or formal elements. The 3 parts are formally joined and function as an integrated unity or whole. The parts are different though and distinguishable from one another.

The spirit part is non material and invisible being formed of spiritual substance or stuff. The Soul likewise is non material. The body is comprised of different kinds of matter or physical substances or stuff. The spirit and soul are distinguishable and structurally different parts - different things.

Non- believers have a spirit as well as a soul, though in an non-believer the spirit part is dead in the sense that it is out of relationship with the Living God.

The Body is seen as the home of the soul and spirit during the individual's life on the earth. We are to be good stewards of our bodies which are a gift from God.

At death the spirit and soul unit detaches from the body which dies and returns to the ground. The spirit and soul remain integrated and go into the intermediate state or hades / sheol (the afterlife) either to Tartarus (the nether gloom) for unbelievers or Paradise (with the Lord) for believers. At resurrection the spirit and soul unit is restored to a resurrection body for purposes of either judgment in Gehenna, the Lake of Fire (the second death), or eternal life.

When an individual is born again the spirit part is restored, renewed, and regenerated and becomes indwelt by the Holy Spirit. The relational function of the spirit between the soul and the presence of God is restored. The difference between the tripartite and trichotomous understandings is that the tripartite understanding does not believe in a separation between the spirit and soul during the history of the human individual. These parts remain integrated though they are formally different.

Trichotomist

from the Greek tricha = threefold or "into three" and temnein = to cut.

Although views of this kind need not presuppose some kind of "irreconcilable antithesis between spirit and body" (Created in God's Image - A. Hoekema p.206) which would be a divisionist view.

Views like this say that the human individual is comprised of 3 different and distinguishable parts or formal elements : a spirit, a soul and a body. More notably they say that during the history of the human being, these parts may be separate from the other parts.

It is the understanding that non-believers in Jesus do not have a spirit part of their being. They believe that the spirit part was lost to humanity when Adam fell. When a person becomes a believer in Jesus and are born again they acquire a spirit part to their beings. So in whatever they define spirit to be, it is something that is deficient in non-believers.

Further, they believe that the spirit and soul as a unit separate from the body at death, though in unbelievers who die, only the soul and body separate and their belief is usually that the soul continues, but some believe that non-believers are annihilated, that is that they cease to exist at death. That is, their body dies and their soul which is dependent on the body ceases to exist because it is not connected to a spirit.

Some hold a divisionist teaching and see an opposition between spirit and body, the kind of spiritual / material dichotomization or antithesis and that the soul bridges the two. It is divisionist teaching that is at the root of any teaching that says the body is bad and evil. It derives from the philosophy that matter is corrupt and that spirit is a higher form and that the two are in opposition. This idea has its source in the Greek philosophy of Plato and others and is not at all natively Christian though it has been imported into Christianity. It was highly influential in the heretical sects of the Gnostics.

General Points

Christian understandings in this area have been muddied by elements imported from philosophies outside Christianity as they have with many other issues. This has confused many issues, but as a positive it has and is forcing Christians to study the scriptures and realize what they believe about these things. Just as the heretic Marcion forced the church to state which books It would accept as scripture and which books it would reject, so God is using ancient greek philosophies and modern academic philosophies to test and refine Christian understandings about these issues.

This issue is also closely connected to the christian's need to define christian spirituality, and it is this need that the church as a whole is only slowly waking up to. The enemy is bringing in more false philosophies and spiritual ones at that. The New Age Movement, Theosophy, Scientific Philosophy, Maryolitary in the Catholic Church and a host of others that are challenges to christians to better understand and articulate christian spirituality.