Fact

A fact is something that is the case: an existent, an attribute, an event or a relationship, taken as it actually is rather than as anyone believes it to be. Objectivism treats facts as the terminus of every cognitive claim, and distinguishes sharply between facts that are metaphysically given and facts that are man-made.

“Existence is Identity, Consciousness is Identification.”
~ Ayn Rand, "Galt's Speech", For The New Intellectual

Facts and truth are not the same

The distinction is easy to blur and worth keeping.

A fact is in reality. The table is oak. That is so whether or not anyone has noticed.
A truth is in a mind. The judgment that the table is oak is true, and it is true because of the fact.

Facts are therefore not true or false. They simply are. What can be true or false is an identification of them. This is why Objectivism says truth is the recognition of reality rather than a property reality has: reality does not correspond to anything, it is what everything else has to correspond to. See Identity and Objectivity.

How facts are reached

Many facts are available by direct observation: a person can look and see that the table is there. Perception is the base, and it does not interpret — it delivers what is there in the form the sense organs make available. See Perception and Senses.

Facts beyond direct observation are reached by proof: establishing a chain of reasoning that connects the claim to what can be perceived. The chain may be long — the mass of a distant star, the cause of a disease — but every step must be one a person can follow, and it must terminate in observation. A claimed fact whose chain does not terminate there has not been established. See Hierarchy and Logic.

A claim offered with no such chain and no evidence at all is arbitrary: not a disputed fact, not a probable one, but outside the classification entirely.

The metaphysically given and the man-made

Objectivism divides facts into two categories, and treats confusing them as a serious error.

Metaphysically given facts are those that result from the nature of existence: the laws of physics, the properties of matter, the fact that men are mortal. They are what they are, could not have been otherwise, and are to be accepted as the absolute they are. The appropriate response is to identify them and act accordingly.
Man-made facts are those that result from human choices: a building, a law, a career, a war. They exist, and so are facts about the world, but they could have been otherwise and can be judged, resisted and changed.

Treating a man-made fact as though it were metaphysically given produces resignation — the assumption that a bad institution is simply how things are. Treating a metaphysically given fact as though it were man-made produces the opposite error: the expectation that reality will yield to wishing. Both are failures to identify what one is looking at. See Metaphysically Given versus the Man Made and Volition.

Common misunderstandings

That "fact" means what has been proved. Proof concerns a person's knowledge. A fact is a fact before anyone establishes it, and remains one if nobody ever does.
That facts are theory-laden and therefore not objective. Identifying a fact requires concepts, and concepts are formed by a mind. What the mind contributes is the form of the identification; the content is supplied by what is there. Denying the first half gives intrinsicism, denying the second gives subjectivism.
That values are not facts. On the Objectivist account they are: that something furthers a given organism's life is a fact about that organism and that something. What is denied is that values are intrinsic — good apart from any beneficiary. See Value and Is-Ought Dichotomy.
That an unfalsifiable claim is a claim about a fact we cannot check. If nothing whatever would bear on it, no fact has been asserted. See Arbitrary.

Discussion

References

Rand, Ayn. "The Metaphysical Versus the Man-Made", Philosophy: Who Needs It. Bobbs-Merrill, 1982.
Rand, Ayn. "Galt's Speech", For The New Intellectual. Random House, 1961.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapters 1 and 5.

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