"For centuries, the battle of morality was fought between those who claimed your life belongs to God and those who claimed it belongs to your neighbours. And no-one came to say that your life belongs to you and that the good is to live it."
~ John Galt, Atlas Shrugged
Justice. Honesty. Independence. Integrity. Productiveness. Pride. Rationality.
Rationality. Man's superlative tool for survival is his mind. Unlike an animal, he does not have instincts that absolve him of the duty of thinking. Man has impulses and whims, but these are not the same as instincts - instincts contain the knowledge of how to act, not just the end goal which should be achieved. To satisfy his needs and whims, man must employ reason. Reason, the cognitive process, involves the collection of data by perception and its processing by a man's precepts and brain. The virtue of Rationality is the recognition that one cannot cheat reality, that the law of identity - A is A - cannot be cheated. An Objectivist must never submit himself to unproved faith - to do so is to make a concession to the irrational, to sabotage his consciousness by accepting that which is not so. An Objectivist should always strive to increase his knowledge and perception of reality, and never to sacrifice this virtue to any other. Man's own mind is his superlative value, and when it is sacrificed so are all other virtues.
The virtue of Justice means that man should never seek or grant the unearned. Sacrifice, the virtue preached by the mystics and the altruists, means the sacrifice of a man's higher values to a lower one. Paying for the education of your child is not sacrifice - paying for the education of a stranger is. Lending money to a beloved friend when you can afford it is not sacrifice - giving to a man who you have never met, or beyond your means, is. An Objectivist should appraise men just as scrupulously as he appraises an inanimate object, with the same respect for reality - to grant approval to a rotter in the same proportion it is granted to a man of supreme goodness is an act of moral fraud. Thus, in dealing with all men, an Objectivist must follow his perception of reality and never to deny that, when it comes to sentient beings, A is still A.
Integrity means the recognition that one's life is one's highest value and that it must never be sacrificed for the whims of others. It must be disposed of as one sees fit as a rational being. Ownership without the right of disposal is no ownership at all - if one allows one's life to be dictated by another, one has sacrificed one's highest value. Integrity means never to sacrifice higher values to lower ones. It means never to grant approval or be complicit in what one knows to be evil. On the issue of moral principles, there can be no compromise. "In any compromise between food and poison, it is only death that can win. In any compromise between good and evil, it is only evil that can profit." Thus an Objectivist must always be true to his own rational principles, not his whims or desires nor the whims and desires of others. An Objectivist must never shy from pronouncing moral judgement on what he knows to be evil.
Honesty is closely related. Just as Rationality is honesty to oneself, the virtue of Honesty is honesty to others. Lying to a man has two effects on him. One is to distort his view of reality, to force him to act against it - for him, A has become non-A. The second follows from this, in that it makes you a slave to his fake reality and his deluded consciousness. To achieve a value by lying and defrauding others is to achieve the unearned - and neither love nor fame nor money are values at all when gained by fraud. To be true to reality when dealing with others is the most profoundly selfish virtue a man can practice, because it lets no sliver of unreality into his life, no fake stories for him to be enslaved by. A man of integrity and rationality is a man of honesty.
The virtue of Productiveness is the recognition that one's life - that highest value - must be sustained by one's own effort. If you were on a desert island, you would have to employ your rationality to sustain your life. Being in the middle of a productive society does not absolve you of the duty of rationality applied to a productive process, nor give you the right to fleece the productive members of society for your own benefit. Productiveness is the constant application of one's mind - one's creativity, one's ambitiousness, one's ingenuity - to shape matter into a form useful to sustaining one's life and achieving happiness. Any man who chooses to substitute a gun for a syllogism and seek to sustain his life by theft, fraud and murder - in any other way than dealing with his fellow men as rational traders - is immoral and evil. No man can expect his life to be sustained by the work of others, and only productiveness allows him to sustain it morally, by himself.
Pride is the recognition that one's life is one's highest value, and that man must strive to earn these qualities of the soul just as he earns his wealth and the sustaining of that life. He is born able to create himself in the image of a rational being who earns his life and earns his soul. Pride is a virtue appropriate to a rational, moral person who knows that his own life is his own highest value, and that the pursuit of his own happiness is the only way to dispose of that life. The altruists preach that one's own life is not one's highest value - it is to be given to others and disposed of by them as they wish. Objectivism is the morality of life, not death - justice, not need. Self-esteem and the happiness that accompanies it are the only proper ends of a man's own life. So-called "brotherly love" is appropriate when it is of value to a man - not when it is extorted at the point of a gun.
Finally, Independence. Independence means never accepting the judgement of another over one's own, never substituting the views of another for your own knowledge of reality. Never putting blind faith in a mystic when what he says contradicts your own knowledge of reality. This is the acceptance that nothing can absolve you of the duty of perception and rationality, nothing can be a middle-man between yourself and existence. A is A, someone else's subjective opinion which disagrees with your own is non-A.
For further information I direct you to the books of Ayn Rand - specifically, Atlas Shrugged, The Fountainhead and The Virtue of Selfishness. For social applications, see Capitalism: the Unknown Ideal.
I use male examples simply to follow Rand's convention and because it avoids confusion. I don't subscribe to monolithic belief systems in their totality, so I don't want to argue with you about it. Go ask Nathaniel Branden.