Primacy of Existence

The primacy of existence is the Objectivist principle that reality exists independently of consciousness, and that the function of a consciousness is to perceive reality rather than to create it. It is not one of the three axioms but a corollary of them, and it is the premise on which the possibility of objectivity rests.

“Existence exists—and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists.”
~ Ayn Rand, "Galt's Speech", Atlas Shrugged

Where it comes from

The principle is not asserted; it is read off the axioms. Existence and consciousness are both axiomatic, but they are not on the same footing, and the relation between them is fixed by what consciousness is.

Consciousness is the faculty of perceiving that which exists. That is its identity, and an identity is not optional. A faculty of awareness that had nothing to be aware of would not be a faculty operating in a void — it would not be a faculty of awareness at all, because there would be nothing for the word "awareness" to name. Awareness is therefore relational: it presupposes something other than itself, which means existence must already be there for consciousness to be consciousness.

So the two axioms are not symmetrical. Existence would be what it is with no one looking. Consciousness could not be what it is with nothing to look at. That asymmetry is the primacy of existence, and its denial is the primacy of consciousness.

What it does and does not claim

The principle is about the relationship between mind and world, not about the power of the mind.

It does not say consciousness is passive or unimportant. Man's mind is his means of survival, and by acting on knowledge he reshapes his surroundings on a scale no other animal approaches. What he cannot do is bypass the facts he is acting on. Bacon's formulation, which Rand endorsed, is the practical statement of the point: nature, to be commanded, must be obeyed.
It does not say consciousness is unreal. Consciousness exists; it is part of reality and has a specific identity. What it does not have is the power to be its own object first, or to legislate what the rest of reality will be.
It does say that wishing, believing, fearing and agreeing change nothing about the object. A fact is not made true by consensus and not made false by discomfort. This is why Objectivism treats the arbitrary as outside the realm of cognition rather than as a claim awaiting refutation.

Consequences

The principle is where several other Objectivist positions are grounded.

Objectivity is possible and is an achievement. If the object is what it is regardless of the observer, then a method exists for conforming one's judgments to it, and logic is that method. Objectivity is neither the intrinsicist's claim of unmediated contact with the object nor the subjectivist's abandonment of the object. See Intrinsicism and Subjectivism.
The metaphysically given versus the man-made. Facts that obtain independently of human choice are to be accepted as absolutes; things men made can be judged, changed or rejected. The distinction only makes sense if there is a class of facts nobody voted on.
Creation of the universe is rejected. A creator would have to be a consciousness aware of nothing prior to what it creates, which the identity of consciousness rules out. Objectivism's case against a creator god runs through this principle rather than through evidentiary argument about any particular religion. See Religion and Mysticism.
Emotions are not tools of cognition. They are effects of one's value premises, so treating them as evidence about the world is a covert appeal to consciousness as primary. See Reason-Emotion Dichotomy and Emotions.

Common misunderstandings

That it means "matter is primary" or that it is a form of materialism. It says nothing about what existence is made of. That is a question for physics, not for metaphysics, and Objectivism does not settle it philosophically. See Mind-Body Dichotomy.
That it means human beings cannot be wrong about anything. The opposite: error is possible precisely because the facts do not adjust themselves to the belief. The principle is what makes error a meaningful notion.
That it denies that our knowledge is shaped by our means of knowing. All awareness is awareness by some means, in some form. Objectivism holds that a form of perception is how something is perceived, not a screen between the perceiver and it. See Senses and Realism.
That the disagreement with Kant is over whether the mind is active. It is not. Both hold the mind does something. The dispute is whether what it does is identify a reality that was already there, or constitute the only reality available to us.

Discussion

References

Rand, Ayn. "Galt's Speech", Atlas Shrugged. Random House, 1957.
Rand, Ayn. "The Metaphysical Versus the Man-Made", Philosophy: Who Needs It. Bobbs-Merrill, 1982.
Rand, Ayn. Introduction to Objectivist Epistemology. The Objectivist, 1967.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 1.

External links

Primacy of Existence vs. Primacy of Consciousness at the Ayn Rand Lexicon

Metaphysics Topics
Axioms: Existence, Consciousness, Identity • Corollary: Causality, Primacy of Existence
Reality: Primacy of Existence, Existents, the Metaphysically Given versus the Man Made