First dualism. Made famous by (though arguably not original to him) Descarte came up with this idea and fleshed out the logical arguments behind how the mind is separate, distinct, and different than the body/brain (for the remainder of this entry when I say 'body' just understand that I'm [more than likely] referring specifically to the brain as the controlling organ of the body). For one of his main arguments he used for dualism Descartes used identity property laws. The essence of the mind is in thought, which isn't a measurable substance. You can't measure the size of an idea or concept. I am thinking of a white elephant right now (bet you are too) and you cannot tell me how big that thought is. It cannot be measured in pounds or inches or any other system of measurement. The human brain (head) is eight pounds (thanks cute kid from Jerry MacGuire) and even the electrochemical impulses in the brain can be measured using electroencephalographs and other tools. Therefore based on the properties of identity, they cannot be the same thing if they are different essential characteristics.
Now materialism. This is a much easier to explain idea because the idea of materialism is simple, the brain is all there is. There is no mind-body problem, there's no such thing as the mind as a distinct thing from the body. The reason we say 'mind' and other mind-related terms, is simply a difference in terminology. Saying, 'mind' is the same as saying 'brain' and saying, "I feel angry." is the same as saying, "there's a certain state of chemicals in my brain." The only dualism is in terminology.
Now, both have their problems, and to be perfectly honest I don't have any answers. I feel that both sides are intractable and cannot offer all the answers. For dualism the primary problem is in the interaction between mind and body. If the mind is intangible then how does or how can it influence the body? Is it a two-way street? Is it a one-way? Is there no interaction? None of these seem possible.
On the other hand, materialism just kills all conception of the mind. As hard as one might try, one cannot get rid of the mind, thoughts cannot be simplified to just chemical processes. Even knowing that one's brain is mainly a complex system of electrochemical reactions to stimuli doesn't make me think of those processes while I'm thinking. It seems to be obvious that thought is beyond just the chemical processes that go on inside your skull.
Really this discussion boils down to atheism and theism. Either there is something more than just the material or there isn't. If you believe there is no such thing as god, then there must be no such thing as the mind/soul/spirit. If you believe there is something more than just the material, then there is some form of mind distinct from the body.
Here's my personal problem in this question, I think it's indubitable that there must be something more than just our bodies. I'm a dualist (I'd say that any theist is and must be), but I have absolutely no idea how the two different parts interact. From theology it's obvious that God is (in some ways) immaterial and spirit, akin to soul or mind, but man isn't God. Now my theological answer is that God has made man in His image in that our souls can interact and influence the material to a limited extent like his Spirit is active in our lives and world. I don't have any better answer than that.