We don't understand all aspects of the consious mind yet, but that by no means makes it an unknowable, mystical, soul-like, ephemeral aspect of ourselves. Not very long ago people assumed that the nature of electricity was so deeply shrouded in mystery that it would forever be beyond our understanding.
The conscious mind is clearly a set of actions performed by the brain. You can prove this by physically affecting the performance of the brain in any of various ways. Many chemicals change the conscious mind; some even extinguish it for a while, like the anaesthetics. Alcohol, hallucinogens, opiates, tranquillizers all alter the way the physical brain operates, and alter the consciousness that is the action the brain performs. A very physical whack on the head alters the way the brain acts too. A surgeon's knife or electrical probe also alter the physical nature or actions of the brain, and thereby, often strikingly, the conscious mind. Every night your consciousness goes through many cycles where it ceases for a little while, and for a time enters the unruly state of self-stimulation we call dreaming.
We have begun to unravel the tapestry that is the conscious mind. It is an amazing invention of nature. But don't be deceived into thinking that it is impossible to understand how it works. Nothing found so far indicates that there will be anything unknowable in this wonderful machine.
About 50 years ago Alan Turing provided a proof that a general purpose computer can simulate anything. Because of this a general purpose computer is sometimes called a Turing machine. It is fairly clear that the brain can be simulated by a Turing machine, and the action it performs will be consciousness. NOTE: twwells pointed out a fundamental mistake I made here. A Turing machine can't simulate anything; it can emulate any other general purpose digital computer. Many ordinary phenomena can be simulated by a digital computer, but it is not necessarily clear that a Turing Machine can simulate the brain. For instance the brain's activity might depend upon analog effects that are highly sensitive to very small initial conditions -- that is, "chaotic". I can't prove it yet, but I'm persuaded that the brain is digital because of the way it goes to great lengths to pull stable results out of biologically "noisy" systems.
I am not saying that it is easy or that it will happen really soon -- but it will definitely happen. Computing power is increasing in capability so rapidly that conservative projections put a $1000 computer in your hand in just 20 years that has the capacity of a human mind. Just another 30 or so years after that a $1000 computer will be equivalent to the processing power of all the human brains on the planet. And that is a conservative projection!! There is much to indicate it could happen well before that.
Now, having the capability to simulate a human mind is clearly not the same thing as actually simulating a human mind. The latter depends upon another technology: scanning. At present we can non-destructively scan the brains of very small animals (insects and such-like) resolving down to the level of individual nerve cells. There are formidable technological hurdles to overcome before we can scan human brains to anything like that level of detail. Worse, we need to be able to scan at a level of detail much higher -- we need to be able to resolve individual receptor sites on synapses. This technology may be some time in coming. Some say that use of nano-machines to travel inside the brain via its blood vessels and map it from within has a much greater chance of success in the near future. Nanotechnology is coming of age very rapidly. I have an open mind on this. I am sure that a way will be found, though, and that it will be found inside the next few decades.
If my brain is simulated with perfect accuracy then the simulation will perform the set of actions that is by any sensible definition my consciousness. If my original biological body is alive at the same time as the simulation is started then momentarily there will be 2 identical versions of me. After that point they will, of course, begin to diverge in experience.
I have no desire to die -- I have too much curiosity and want to do too much. If I lived several hundred years I would achieve only a fraction of what I want to do and learn. But my biological body comes with an expiry date. The only way I can get around that is to copy my brain's physical substrate as faithfully as possible to some effectively immortal format and have my consciousness continue there.
Computers' power requirements are decreasing rapidly. I don't expect it to be long before computers need just ambient energy -- light, heat, sound in order to power their functions. Already my Palm computer's tiny rechargeable batteries enable it to run on a trickle of power for up to a month between rechargings.
I expect that sometime in the next 50 years we will have the ability to copy or backup our brains into some form of computer. They won't look much like today's computers and they will have negligible power requirements. You can make philosophical arguments that a backed-up mind is not exactly the same as the original, but in the final analysis it is practicality that matters. I am not exactly the same as I was last year, but I am close enough. I have the memories that give me continuity even though I actually exist in a series of nightly-daily stop-start events. The backed-up version of me will have my memories and my personality and the slight time gap between copy-time and restart-time will be unimportant. Close enough is good enough and will enable me to exceed my biological body's use-by date. I think in practice this is all that will matter. Talking to someone over the phone is in some way talking to "just" an electronic copy of the voice, but we all know that in practice there is a real person at the other end. So it will be with backed-up minds. We can argue that they are just fake brains, but in reality everyone will know that there will be genuine consciousnesses inside them.
But I know you will not change your mind on this. Nobody ever does. The only way will be to see with your own experience. Hopefully I will be able to show you in a couple of decades.