Rule of Fundamentality

The rule of fundamentality is the Objectivist standard for what a definition must name: not any characteristic that happens to distinguish the units of a concept, but the fundamental one — the characteristic on which the greatest number of the others depend.

“Metaphysically, a fundamental characteristic is that distinctive characteristic which makes the greatest number of others possible; epistemologically, it is the one that explains the greatest number of others.”
~ Ayn Rand, "Definitions", Introduction to Objectivist Epistemology

The problem it solves

Any group of things can be distinguished from everything else in many different ways. Man can be defined as the rational animal, the tool-making animal, the featherless biped, the animal that laughs, or the only creature that cooks its food. Every one of these picks out exactly the same units.

If definitions were merely a way of identifying which things we are talking about, all of these would be equally good and the choice would be arbitrary — which is the nominalist conclusion. The rule of fundamentality is what makes the choice objective without making it intrinsic:

Rationality is what makes the rest possible. Tool-making, cooking, laughter, language and every other human distinctive follow from a conceptual faculty. Nothing follows from featherlessness.
A definition therefore condenses a body of knowledge. Naming the fundamental characteristic implicitly carries everything that depends on it. This is why definitions serve unit-economy rather than merely labelling.
The wrong choice breaks the hierarchy. Define man by an incidental trait and the concept still picks out the right things while explaining none of them, which is how a term becomes a floating abstraction while remaining in correct use.

Rand's formulation gives the rule two faces because causation and explanation run in opposite directions: metaphysically the fundamental characteristic is the cause of the others, epistemologically it is what accounts for them.

Definitions are contextual, not arbitrary

The rule does not imply that a definition, once correct, is correct forever. Which characteristic is fundamental depends on the context of knowledge available.

A child's first definition of man might be "the thing that walks on two legs and talks". That is not a mistake at his level — it names what is fundamental among the facts he has. As his knowledge widens, the definition is revised, not because the earlier one was false but because more is now known about what depends on what. See Certainty and Objectivity.

This is Objectivism's answer to two standard positions at once. Against intrinsicism, a definition is not a report of an eternal essence residing in the thing. Against nominalism and conventionalism, it is not a stipulation that could have gone any other way. It is an objective identification made within a context. See Analytic-synthetic dichotomy.

Where the rule gets used

Detecting package deals. Ask what characteristic the alleged units share fundamentally. If the honest answer is "they are often found together", the grouping fails the rule.
Detecting invalid concepts. A term whose units have no fundamental common characteristic has nothing for a definition to name.
Settling substantive disputes. Arguments about whether something "really is" capitalism, or a right, or a sacrifice are frequently disputes about which characteristic is fundamental — which is a question about reality, and answerable.

Common misunderstandings

That it means finding the most important characteristic. Importance is relative to a purpose. Fundamentality is a matter of dependence: what makes or explains the rest.
That the definition must state everything known about the units. It states the fundamental characteristic and the genus. Everything else is implicit in the concept, not packed into the formula.
That contextual definitions are relativism. A definition is correct or incorrect given a context of knowledge, and contexts are not interchangeable. Wider knowledge subsumes narrower knowledge rather than contradicting it.
That a good definition settles a question by itself. It records a conclusion reached by examining the units. Arguing from a definition one has not established is rationalism.

Discussion

References

Rand, Ayn. "Definitions", Introduction to Objectivist Epistemology. 2nd edition, Meridian, 1990.
Peikoff, Leonard. "The Analytic-Synthetic Dichotomy", in the same volume.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 3.