The word "Tao" (Usually pronounced "Dow". I don't know why. But some pronounce it with the "t".) simply means "the way". Taoism speaks of the way of the universe, the way of nature. It speaks of what it considers the best way to live. That is, living without anger, hatred, frustration, and all the other negative emotions. Living as the universe exists, without effort or worry.(1)
And how is that accomplished? In Taoism, it all comes down to this: Give up desire. The Tao Te Ching says:

If people lack knowledge and desire, then intellectuals will not try to interfere. If nothing is done, then all will be well. (Verse 3)
Observers of the Tao do not seek fulfillment. Not seeking fulfillment, they are not swayed by desire for change. (Verse 15)
Without desire there is tranquility. And in this way all things would be at peace. (Verse 37)

That�s all you need to know. Such a simple thing, really. Give up desire, and you will be content. And in your contentment, you will be able to find happiness.
I could stop now. And if you followed that advice, all would become clear to you. But I'll explain the nuts and bolts of it all.
In the Tao, everything acts only within its nature. The sun burns, the hawk hunts, the water runs, the tree grows. Water does not desire to run uphill. It does not attempt to act in a manner inconsistent with its nature, the Tao. The hawk does not desire to burrow into the ground as the mole does. It does not attempt to fight the Tao.
And despite the fact that things act only according to their nature, every single thing that is necessary for the continued existence of the universe is accomplished. My very favorite passage from the Tao Te Ching: "Tao abides in non-action, yet nothing is left undone." (Verse 37) Things run perfectly without thinking and planning, without fighting against the universe.
The problem is that we are no longer part of the Tao. We have lost our Way. We don't see the glorious harmony of it all, the perfection of the universe's intricacies. And we don't even know that we are lost! We have separated ourselves from everything, and then go back and try to possess it all. That damned desire!
We desire things that the universe does not naturally give us. If the earth was made entirely of gold, we would not desire gold. We don't desire what we have, or can have whenever we want. We only desire what is not readily available to us.
Therefore, when you act to attain what you desire, you are fighting the Tao. You are fighting the natural order of the universe. That's worth repeating: You are fighting the natural order of the universe! Doesn't that seem like a strange, arrogant, and impossible thing to do?
The problem is that desire simply can't be satisfied. On the practical side, it just doesn�t work. Yes, many individual desires can be achieved. But as soon as you get one thing that you desire, another pops up. Then another, and another..... Eventually, you will desire something that you can't have. Maybe you don�t have enough money. Maybe not enough time. Eventually, you will be frustrated. You may envy those who have what you cannot. You may hate them, steal from them, or kill them to take what they have.
On the spiritual side, desire is all consuming. Even if you could eventually get any particular thing that you set your sights on, there is simply no end to the wanting. Desire itself cannot be satisfied. It�s an all-or-nothing type of thing. If you have it, there is no end. No point where you say, "Ah, I now have everything I desire. I can relax now and enjoy all that I have." Give it up completely, or be prepared to spend your entire life trying, and often failing, to get one thing after another after another.(2)
So give up desire, and give up the negative results. The constant wanting, frustration, and anger. Accept what the Tao provides you, desire nothing that is not provided, and you will live in peace and happiness. Do only what is necessary to live, to eat, to breath, to be. If the Tao does not provide it to you, you don't need it.
This might seem somehow wrong. After all, doesn�t giving up desire mean giving up caring and being happy with something? Not really. Someone could taste chocolate for the first time in their life at the age of 40, and absolutely love it. But never having tried it before, their enjoyment of it clearly had nothing to do with desire for it. We will like and dislike things even if we are free of desire. Our preferences will still be there. The thing is, if you're not spending so much time and energy trying to satisfy desires, and being frustrated and angry over those desires that you can't manage to fulfill, you will realize that there are a thousand things every day that will make you happy. Pay attention to the moment. Notice the beauty and happiness that is around all the time. That old �stop and smell the roses� idea. Just because I enjoyed something, say a particular meal, doesn�t mean I have to drive myself crazy wanting to have it the next day. If I have it again, I�m happy again. But I can still be happy with other meals that come my way. Here�s a great story to illustrate my point.(3)

A man was being chased by a ravenous tiger. He came to the edge of a cliff and began to climb down a hanging vine. Then he looked and saw a second, equally ravenous tiger waiting at the bottom. At that moment, a mouse began to gnaw at the vine. Something caught the man�s eye - a luscious, red strawberry growing just within his reach. He plucked it and ate it and exclaimed, �How delicious this is!�

This is obviously an extreme example of giving up desire; giving up the desire to continue living another five minutes. But when he realized that he could not change his fate, he let go of the desire to do so, and took the pleasure that was available to him. He was not thinking, "I want a strawberry before I die." He just happened to find one, and ate it. Just because he was not going to experience pleasure beyond the next five minutes, doesn�t mean that he couldn�t experience it within the next five minutes. Holding on to his desire to live would have precluded his ability to be happy in the last moments of his life. Remember the story when something infinitely less important than your imminent demise is bothering you, and realize that it�s probably not that important anyway. And if it can�t be changed, it doesn�t matter how important it is. Let it go.

The natural consequence of having no desire is something called wu-wei. This is usually translated as things like "non-action", "non-contrived", "non-ado", etc. The best explanation of the term that I've seen is from The Tao of Zen, by Ray Grigg. He says, "When non-doing appears as inaction it is peaceful, silent, and still; when it appears as action it is thoughtless, reflexive, and intuitive." When we desire, our actions are planned and schemed. They are for a reason, with a goal in mind. Our energy is wasted trying to change circumstances, fighting the natural order of the universe. But when we practice wu-wei, our actions are unmotivated and instinctual. They are natural reactions to the moment.
Wu-wei is not gained through any desire, effort, or plan. You do not say, "I will achieve wu-wei in the following manner..." It is simply the way things without desire act. If you desire nothing, your actions will not be the result of any intent to satisfy a desire. Instead, your actions will simply be in accordance with what is happening around you. Because of our self-preservation instinct, the man in the story above did what he could to survive. His actions were the response to the moment; needs and instinct were driving him. (Of course, most of us desire to live, and his actions might be seen as attempts to achieve this desire. But animals that never think "I want to live" behave the same way. Without any conscious desire to live, the man would have instinctively acted the same way.) When he knew that his life was over, when there was no spontaneous/instinctual act left to perform (and, if desire to live did, indeed, play a role in his previous actions, he realized that no possible action could achieve this desire, and so stopped acting on it), he ate the strawberry. This was not a planned action based on any desire. It was a spontaneous action, made possible by the situation he was in.

OK, that's the message of Taoism. But, since I enjoy such things, I'm going a step farther. Here is my Psychological Commentary on Taoism. This won't increase your understanding of Taoism (or perhaps "your understanding of my interpretation of Taoism" is more appropriate), so don't worry if you don't feel like reading it. I just wanted to say why I think we are removed from the Tao, and what I think our relationship to the Tao should be. I quote Jung, and fun stuff like that :)

And here are a bunch of quotes. They deal with the concept of giving up desire, accepting what comes to you through the Tao, and not fighting against it. These are all from non-Taoist sources, which are, nevertheless, clearly expressing this ideal.

Black Elk-
It is from understanding that power comes; and the power in the ceremony was in understanding what it meant; for nothing can live well except in a manner that is suited to the way the sacred Power of the World lives and moves.

In the March 1999 National Geographic, page 24, Nouhou Agah says:
I will tell you something about the Sahara. This desert is very simple to survive in. You must only admit there is something on Earth larger than you...the wind...the dryness...the distance...the Sahara. You accept that, and everything is fine. The desert will provide. Inshallah. If you do not, the desert will break you. Admit your weakness to the Sahara�s face, and all is fine.

Jesus� Sermon on the Mount-
Matthew 6: 25-30
Therefore I tell you, do not be anxious about your life, what you shall eat or what you shall drink, nor about your body, what you shall put on. Is not life more than food, and the body more than clothing? Look at the birds of the air: they neither sow nor reap nor gather into barns, and yet your heavenly Father feeds them. Are you not of more value than they? And which of you by being anxious can add one cubit to his span of life? And why are you anxious about clothing? Consider the lilies of the field, how they grow; they neither toil nor spin; yet I tell you, even Solomon in all his glory was not arrayed like one of these. But if God so clothes the grass of the field, which today is alive and tomorrow is thrown into the oven, will he not much more clothe you, O men of little faith?

T.S. Eliot:
The state of being a Christian: A condition of complete simplicity (Costing nothing less than everything)

The Rolling Stones:
You can't always get what you want.
But if you try sometime,
you just might find
you get what you need.

From Stranger in a Strange Land, by Robert Heinlein:
The Man from Mars sat down when Jill left. He did not pick up the picture book but simply waited in a fashion which may be described as �patient� only because human language does not embrace Martian attitudes. He held still with quiet happiness because his brother had said that he would return. He was prepared to wait, without moving, without doing anything, for several years.

From Tony Hillerman�s Listening Woman:
The Hopis had held a rain dance Sunday, calling on the clouds - their ancestors - to restore the water blessing to the land. Perhaps the kachinas had listened to their Hopi children. Perhaps not. It was not a Navajo concept, this idea of adjusting nature to human needs. The Navajo adjusted himself to remain in harmony with the universe. When nature withheld the rain, the Navajo sought the pattern of this phenomenon - as he sought the pattern of all things - to find its beauty and live in harmony with it.

Rabbit, in Sing A Song With Pooh Bear:
Harvest what you grow
There�ll be so much to show
And you will have everything you need
(Like you need anything else)

Chapter 10 of The Little Prince, by Antoine de Saint Exupery, is a lesson in Taoism. He is obviously familiar with the Tao Te Ching. Very nice work.

And these quotes express other Taoist ideals-

From The Zen Teaching of Bodhidharma, translated by Red Pine:
To have a body is to suffer. Does anyone with a body know peace?
is similar to Verse 13 of the Tao Te Ching:
Misfortune comes from having a body. Without a body, how could there be misfortune?

This part of Chapter 8 of the Dhammapada:
One who conquers himself is greater than another who conquers a thousand times a thousand men on the battlefield. Be victorious over yourself and not over others. When you attain victory over yourself, not even the gods can turn it into defeat.
reflects this line from Verse 33 of the Tao Te Ching:
He who conquers men has force; he who conquers himself is truly strong.

In Conversations With God, by Neale Donald Walsch, God says:
Within the realm of sublime relationships nothing which exists has an opposite. All Is One, and everything progresses from one to the other in a never-ending circle.
This is very much like this passage from the Chuang Tsu:
When there is no more separation between "this" and "that," it is called the still-point of Tao. At the still-point in the center of the circle one can see the infinite in all things. Right is infinite; wrong is also infinite. Therefore it is said, "Behold the light beyond right and wrong."

This passage is from Zen in the Art of Archery, by Eugen Herrigel:
... the mind or spirit is present everywhere, because it is nowhere attached to any particular place.
It is very similar to Verse 7 of the Tao Te Ching, translated by Feng & English:
The sage is detached, thus at one with all.

The great Hindu text, the Bhagavad Gita, 3/10 says:
Through selfless service, you will always be fruitful and find the fulfillment of your desires.
This is very similar to the next line of Verse 7 from the Tao Te Ching:
Through selfless action, he attains fulfillment.

Actually, just read the Bhagavad Gita, and the Upanishads, and you'll find another million Taoist quotes.

It seems to me that all of these quotes show that Taoism is the core of just about every other religion. Most religions have added religious beliefs to the simplicity of Taoism. Not intentionally, I'm sure. It's entirely possible that the founders of the world's religions had never heard of Taoism. My thought is that Taoism is the most basic teachings of humanity - the Way life must be lived if you want contentment and happiness. Most religions also want contentment and happiness. So, no matter what other ideas they have, they must include these core principles, because you can't give people contentment and happiness without them. Many other beliefs can work fine in conjunction with the Taoist ideals, but they can't do their job without the Taoist ideals. Giving up desire; living selflessly, for the benefit of others; recognizing opposing aspect in all things; etc. All of these things are at the core of the world's major religions.
Here are some other aspects of Taoism that I think are important:
How does a Taoist behave?
Taoist Texts
Zen Buddhism
(1) I should note that I am really trying to be a follower of the Tao Te Ching. Other Taoist texts, such as the Chuang Tzu, may have much in common with it, but there are also some differences; and even points that I see as being in opposition. The Tao Te Ching is what I know, and love, best.

(2) In the introduction to his translation of the Upanishads, Eknath Easwaran explains the never-ending nature of desire like this, referring to the Hindu belief that our true Self, the Atman, is actually the Brahman, the ultimate reality, their conception of God:
The infinite � free, unbounded, full of joy � is our native state. We have fallen from that state and seek it everywhere: every human activity is an attempt to fill this void. But as long as we try to fill it from outside ourselves, we are making demands on life which life cannot fulfill. Finite things can never appease an infinite hunger. Nothing can satisfy us but reunion with our real Self, which the Upanishads say is sat-chit-ananda: absolute reality, pure awareness, unconditioned joy.

I'm not a Hindu. But I won't rule out the possibility that some of the basic ideas of Hinduism are true until I put the time and effort into the practices that they claim will definitely prove that they are true.

(3) I read it in a comic book, and the writer said that this was his favorite Zen story. Zen and Taoism have much in common, and this story works just as well for Taoism. For more on the similarity of Taoism and Zen, see the "Zen Buddhism" link just above.