Rights

Rights are moral principles defining a person's freedom of action in a social context. In Objectivism a right is not a claim on other people's effort but a sanction to act without their interference, and there is fundamentally only one: the right to life, from which all others derive.

“A "right" is a moral principle defining and sanctioning a man's freedom of action in a social context.”
~ Ayn Rand, "Man's Rights", The Virtue of Selfishness

Where rights come from

Rights are not granted by government, by society, or by a supernatural authority. They are derived from the requirements of man's survival, which is why Objectivism treats politics as resting on ethics rather than standing on its own.

The derivation runs:

Man survives by reason, by grasping facts and acting on his conclusions.
A mind cannot be forced to think. Physical force can compel a body, extract a signature, or silence speech, but it cannot make a person understand. Force and thought are alternatives.
Therefore what a rational being needs from other people is precisely that they not initiate force against him.
Rights are the statement of that requirement in social terms. See Initiation of force.

Because the argument turns on the nature of a rational being, rights belong to individuals. A group has no rights beyond those of the individuals composing it, and "the public" is a number of persons, each an end in himself, rather than an entity with claims of its own. See Collectivism and Individualism.

Rights are negative

Every genuine right imposes on others only the obligation to abstain. The right to life is a right to act to sustain one's own life, not a right to be sustained. The right to property is a right to earn, keep, use and dispose of what one produces, not a right to be given property. Free speech is a right to speak at one's own expense, not a right to an audience or a platform.

“The right to life is the source of all rights — and the right to property is their only implementation.”
~ Ayn Rand, "Man's Rights", The Virtue of Selfishness

Property matters here because a right to act is empty without a right to the material results of acting. A person who may think but may not keep what his thinking produces has been left the form of a right without its substance.

"Rights" that are not rights

A claim to a good or service that someone else must provide is not a right, because it can be satisfied only by placing an obligation on another person to serve. Two such claims cannot both be honoured when they conflict, which is a symptom that they were never rights to begin with. Rand argued that a right to medical care, housing or employment means a right to compel a doctor, a builder or an employer, and that this inverts the concept.

Rand and Leonard Peikoff also criticised extensions of the concept beyond rational beings or to entities that are not individuals, including animal rights and collective or group rights. The common error is dropping the context that gave rise to the concept: rights answer the question of how beings who survive by reason must deal with one another.

Rights and government

Rights are moral principles; they do not enforce themselves. The sole proper function of government is to protect them, which it does by holding a monopoly on retaliatory force restrained by objective law: police against criminals, courts for disputes and contracts, and a military against foreign aggressors.

This is why Objectivism rejects anarchism. Leaving retaliation to private discretion does not abolish force; it removes the objective rules governing its use. It equally rejects unlimited democracy: a majority vote does not create a right to violate a minority's rights, which is what a constitution exists to prevent.

Common misunderstandings

That rights come from the Constitution. A constitution recognises and limits government against pre-existing rights. If rights were created by law they could be repealed by law, which is precisely the premise Objectivism denies.
That a right means whatever is legal. Legality and rights are different questions. Slavery was legal.
That "freedom of speech" protects you from other people's reactions. It bars the government from silencing you by force. Private individuals refusing to associate with you are exercising their own rights.
That rights conflict. Properly defined, they cannot, because each is a right to act rather than a claim on others. Apparent conflicts usually involve a claimed "right" of the second kind.

Discussion

Intellectual property
John Locke vs Thomas Jefferson
Veganism under Objectivism — on extending rights to animals
Why theft is neither ethical nor practical

References

Rand, Ayn. "Man's Rights" and "The Nature of Government", The Virtue of Selfishness. New American Library, 1964. Both essays also appear in Capitalism: The Unknown Ideal.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 10.

External links

Individual Rights at the Ayn Rand Lexicon