Rationalism

Rationalism is the epistemological method of treating abstractions as self-sufficient — deducing conclusions from definitions and axioms without checking them against the concretes the abstractions were formed from. Despite the name it is not rationality, and Objectivism treats it as one of the two standing failure modes of a conceptual mind, the other being empiricism.

How the method works

A rationalist begins from a premise he takes to be self-evident and proceeds by deduction, and the appeal of the method is that each step feels rigorous. What it omits is the connection to reality at every point after the first:

The starting premise is not reduced. Its terms are accepted as understood rather than traced back down the hierarchy to the perceptual level. If any of them is a floating abstraction, everything deduced from it floats too.
Concretes are not consulted. A conclusion is checked for consistency with the system, not for whether it describes anything. Examples appear, if at all, as illustrations chosen after the fact.
Context is discarded. A principle is treated as holding in every circumstance whatever, because the conditions that gave rise to it were never identified.
Integration is skipped. A new conclusion is not weighed against the rest of what the person knows, so contradictions between his abstract commitments and his ordinary judgments can persist indefinitely without either being disturbed.

The characteristic result is a person who is internally consistent, articulate, capable of defending his position at length, and wrong in a way that no fact can reach — because on his method no fact was ever admitted as relevant.

Its symptoms

Oscillation between certainty and collapse. Because his conclusions have no anchor beyond the chain that produced them, a single unanswered objection can put the whole structure in doubt. Rationalists are often dogmatic and easily unsettled at once.
Inability to apply a principle to a case. Having never connected the abstraction to concretes, the rationalist cannot recognise an instance of it. In practice he ends up handling each new situation as unprecedented — which is to say, as an empiricist. The two errors meet.
A taste for systems and comprehensiveness. The demand that everything follow from a single premise is treated as a virtue of the theory rather than as a question about reality.
Arguing from definitions. Settling a substantive question by appeal to what a word "really means", rather than by examining the things the word refers to. See Definitions and Analytic-synthetic dichotomy.

Rationalism also tends toward intrinsicism, since a truth arrived at without reference to evidence is easily taken to be simply given, independent of any mind's method of reaching it.

The Objectivist alternative

Objectivism holds that abstractions are formed from concretes and remain accountable to them. See Concept formation and Introduction to Objectivist Epistemology.

The corrective is a two-way discipline rather than a preference for one direction:

Reduction — tracing a concept or principle back down the hierarchy to the perceptual evidence that gave rise to it, which is the answer to rationalism.
Induction — moving from observed concretes to principles, which is the answer to empiricism.
Integration — checking each new conclusion against everything else one holds, so that a contradiction is treated as a signal of error rather than tolerated. See Logic and Objectivity.

Deduction remains valid and necessary. What Objectivism denies is that deduction can do the whole job, or that a premise may be exempted from having a base.

Common misunderstandings

That it means valuing reason. The name is historical and misleading. Rationality is the policy of accepting reason as one's guide, which includes keeping abstractions tied to what they abstract from. See Rationality.
That it is the same as the historical rationalist school. The philosophers usually grouped under that label held that some knowledge is independent of experience. Objectivism's use of the term is broader: it names the method, wherever it appears, including in people who consider themselves empiricists.
That being systematic is rationalistic. Objectivism is itself a system, and holds that knowledge is hierarchical and integrated. The error is not organisation but organisation detached from evidence.
That Objectivists are immune to it. Peikoff's treatment of the subject was directed largely at students of Objectivism, who can learn the conclusions as a set of propositions and never perform the reductions that would make them knowledge.

Discussion

References

Rand, Ayn. Introduction to Objectivist Epistemology. 2nd edition, Meridian, 1990.
Peikoff, Leonard. "The Analytic-Synthetic Dichotomy", in the same volume.
Peikoff, Leonard. "Understanding Objectivism" lecture course, 1983 — the extended treatment of rationalism as a psycho-epistemology.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 5.