In asking such questions, you're touching upon the central problems in the Philosophy of Mind: what is the difference between the mind and the brain?; and what is consciousness?
These are extremely deep questions. You seem to be heading in the direction of Cartesian dualism: Descartes created an ontology (a way of partitioning reality) that divided the universe into physical and mental substances. Objects in the world, including your body and brain, are physical; by their nature they have such properties as extension (they take up space). Your mind (or soul, as Descartes would have it) is mental; it is opposite in nature to physical things, and lacks such properties as extension.
There's more to his theory than that, but that's the important point: divide reality into the physical and the mental. The main problem with this theory is it's theoretically impossible for mental things to influence physical things, or vice-versa. Descartes fudged this horribly, by locating the point of interaction in the pineal gland -- essentially allowing this tiny part of the brain to become a magic box.

These ideas often lead people towards the paradoxical notion of a homunculus [
http://en.wikipedia.org/wiki/Homunculus#Homunculus_argument_or_fallacy_in_the_philosophy_of_mind ]: a "little man" who sits inside the brain, rather like the old Numbskulls cartoon from the Beano [
http://www.fustar.info/2006/12/02/193/ ].
In more modern theory, the questions are more concerned with how mental events (thoughts, consciousness) arise from the physical brain. Although neuroscience can correlate brain areas with certain functions, consciousness is particularly elusive. It's hard to point to a part of the brain and say, "this is consciousness", or "this is a thought".
I believe that consciousness is most likely an emergent entity. Emergence is a fascinating concept that explains how systems with a large number of individually "stupid" parts can develop astonishingly complex behaviour. Such systems are often described as "self-organising". Important examples of self-organising systems include slime mould and ant-colonies. Ant colonies in particular have been demonstrated to develop long-term memories and to solve geometric problems with great precision; and yet the individual ants are far too stupid to be capable of such feats.
Emergence is a new kind of etiology (
http://en.wikipedia.org/wiki/Etiology ), completely divorced from traditional philosophical thought. I believe that the human mind is an emergent system: thoughts and consciousness emerge from interactions (the firing of neurons) within the physical brain.
In my view, the whole brain vs. mind problem is based upon the fallacy of division (
http://en.wikipedia.org/wiki/Fallacy_of_division ). This fallacy insists that any property possessed by an object must also be possessed by its components. Normally we can detect such fallacies easily (see the examples on that link!); but when it comes to our own minds, we seem to have a peculiar logical blind spot.