Virtues

Virtues are the actions by which a person gains and keeps his values. Objectivism names seven of them — rationality, honesty, integrity, independence, justice, productiveness and pride — and treats each as a consequence of holding man's life as the standard of value rather than as a rule handed down from outside.

“Value is that which one acts to gain and keep, virtue is the act by which one gains and keeps it.”
~ Ayn Rand, "The Objectivist Ethics", The Virtue of Selfishness

Values and virtues are two halves of one thing

A value is an end; a virtue is the means. The pairing is not incidental to the Objectivist ethics but its structure. Because man's life is the standard of value, and because man survives by reason rather than by instinct, the question "what should I do?" has a factual answer: whatever the nature of a rational being's survival actually requires.

Rand organised the answer around three cardinal values and their corresponding virtues:

Reason — man's only means of knowledge — is gained and kept by the virtue of rationality.
Purpose — the choice of goals that a life is spent achieving — is gained and kept by the virtue of productiveness.
Self-esteem — the conviction that one is competent to live and worthy of happiness — is gained and kept by the virtue of pride.

The remaining four virtues are the application of rationality to particular kinds of situation: honesty to facts one is tempted to fake, integrity to one's own convictions under pressure, independence to the relation between one's judgment and other people's, and justice to the evaluation of other people themselves.

Why these are not arbitrary rules

Most codes of morality present virtues as duties: things a person owes to God, to society, or to an unexplained categorical imperative. Objectivism rejects the notion of unchosen duty entirely. See Is-Ought Dichotomy.

The derivation runs the other way. Man's survival is not automatic; it depends on identifying reality correctly and acting on the identification. Every one of the seven virtues names a specific way that identification can be sustained or betrayed:

Faking a fact does not change the fact, so honesty is not a favour done to others but a condition of one's own contact with reality.
A conviction abandoned under social pressure was not held as knowledge, so integrity is the difference between having a mind and reciting one.
A judgment borrowed from others is not knowledge either, so independence is not aloofness but the only way a person can know anything at all.

This is why the virtues are described as selfish in Rand's sense. They are not constraints imposed on egoism from outside; they are what practising it successfully consists of.

Common misunderstandings

That the list is a checklist of behaviours. Each virtue names a principle, and applying a principle to a concrete case requires judgment about the full context. Rand's essay "How Does One Lead a Rational Life in an Irrational Society?" is largely about the difficulty of that application.
That the seven are separate and can be traded off. They are aspects of one thing, rationality, so a genuine conflict among them signals an error in identifying the situation rather than a real dilemma.
That virtue is its own reward. Objectivism holds the opposite: virtue is a means to values, and a "virtue" that achieved nothing would not be one. See Sacrifice.
That other traits — benevolence, courage, temperance — are excluded. They are treated as derivative rather than absent. Courage and honesty, for instance, are the same policy under different degrees of pressure.

Discussion

References

Rand, Ayn. "The Objectivist Ethics", The Virtue of Selfishness. New American Library, 1964.
Rand, Ayn. "How Does One Lead a Rational Life in an Irrational Society?", The Virtue of Selfishness.
Rand, Ayn. Atlas Shrugged — Galt's speech, where the seven virtues are stated in sequence.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 8.

External links

Virtue at the Ayn Rand Lexicon