Skepticism

Skepticism is the doctrine that certainty is unavailable to man — that knowledge is at best provisional, at worst impossible, because the human means of knowing is inherently unreliable. Objectivism rejects it, and holds that the doubt it recommends is not a cautious attitude but a self-eliminating claim.

“To arrive at a contradiction is to confess an error in one's thinking; to maintain a contradiction is to abdicate one's mind and to evict oneself from the realm of reality.”
~ Ayn Rand, Atlas Shrugged

The self-refutation

The standard Objectivist answer is that skepticism cannot be stated without being violated. "Nothing can be known" is offered as something known. "All certainty is unwarranted" is asserted with certainty. "The senses cannot be trusted" is argued from evidence that reached the arguer through his senses.

This is not a debating trick but an instance of the stolen concept fallacy: the skeptic uses concepts — knowledge, truth, evidence, proof — while denying the foundations that give those concepts meaning. See Axioms and Truth.

The consistent skeptic has only one move left, which is to stop asserting anything. Objectivism's reply is that this is not a philosophical position but the absence of one, and that it settles no question and guides no action.

The demand behind the doubt

Objectivism's more interesting claim is that skepticism is not the opposite of intrinsicism but its disappointed remainder. The skeptic typically retains an intrinsicist standard of what knowledge would have to be, finds that human cognition does not meet it, and concludes that knowledge is impossible. The standard is usually one or more of:

Omniscience. Unless a conclusion accounts for everything, including what nobody has yet discovered, it does not count. But knowledge is contextual: a conclusion is warranted by the evidence available, and later evidence extends it rather than retroactively invalidating it. See Certainty.
Infallibility. Unless a faculty is incapable of error, its results are worthless. Objectivism replies that error is possible precisely because knowledge requires an active method — and that a method can be checked, which is what logic is for.
Direct access without a means. Unless the mind grasps reality with no processing whatever, what it grasps is "only appearance". This treats the identity of consciousness as a defect. A faculty must work in some particular way to work at all; that it has a nature does not make its results unreal. See Perception and Objectivity.

Each of these is a demand that man be something other than man. Failing it establishes nothing about knowledge — only about the standard.

Varieties

Perceptual skepticism — the senses deceive us, so the world may be other than it appears. See Senses and Realism.
Conceptual skepticism — abstractions are arbitrary groupings, so classification carries no knowledge. See Nominalism.
Inductive skepticism — no number of observed cases licenses a general conclusion. See Induction and Causality.
Moral skepticism — no value judgment can be objectively grounded, which arrives at the same place as subjectivism.

Objectivism treats all four as sharing the intrinsicist premise above, which is why it answers them by reexamining what knowledge is rather than by trying to meet the skeptic's standard.

Common misunderstandings

That skepticism is the same as open-mindedness. Withholding assent until there is evidence is rationality. Holding that evidence could never suffice is skepticism, and it is not a stricter version of the first — it is the abandonment of the standard the first depends on.
That rejecting skepticism means claiming to be always right. Objectivism holds that certainty is contextual and that error is possible and correctable. What it denies is that the possibility of error makes all conclusions equal.
That doubt is free. Objectivism treats a claim advanced without evidence as arbitrary — neither true nor false but outside cognition entirely, to be dismissed rather than weighed. A doubt with nothing behind it is in the same category.
That it is a harmless academic position. Rand's charge was practical: a culture convinced that nothing can be known cannot defend anything, and vacates the field to whoever is willing to assert. See Faith and Mysticism.

Discussion

References

Rand, Ayn. Introduction to Objectivist Epistemology. 2nd edition, Meridian, 1990.
Rand, Ayn. "Philosophical Detection", Philosophy: Who Needs It.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 5 — certainty, the arbitrary, and the answer to skepticism.