Nominalism

Nominalism is the doctrine that universals are only names — that concepts such as "man" or "table" pick out no real similarity among the things they subsume, but are labels attached to arbitrary groupings by convention or resemblance. Objectivism rejects it as the subjectivist answer to the problem of universals, and holds that it makes conceptual knowledge impossible.

The problem it answers

Everything a person perceives is a particular; almost everything he knows is held in abstractions that name no particular. What do those abstractions refer to? See Introduction to Objectivist Epistemology.

The nominalist answer is: nothing beyond the word. There is no attribute the members of a class really share, only a decision to call them by one name. Its usual grounds:

Only concretes are perceived. Nobody has ever seen "man" as such, only individual men — so "man" cannot correspond to anything observed. This is empiricism taken to its conclusion.
Classification could have been done differently. Since any two things resemble each other in some respect, which resemblances get a name is a matter of convenience.
Definitions are stipulations. If a word means what we agree it means, then disputes about what something "really is" are disputes about usage.

Why Objectivism rejects it

It cannot account for the groupings it admits. A nominalist has to explain why "man" works and "all blue things and Tuesdays" does not. Any answer he gives — that the first is useful, natural, or based on resemblance — concedes that something in reality is doing the selecting.
"Resemblance" is the concession. If concretes genuinely resemble one another, the resemblance is a fact about them, not about the observer. Objectivism's account is that the resemblance is real and quantitative: similar things differ in the measurements of a shared attribute, and a concept is formed by omitting those measurements while retaining the attribute. See Measurement omission and Conceptual Common Denominator.
It destroys knowledge along with concepts. If classification is arbitrary, then every general statement is a statement about our vocabulary rather than about the world, and science becomes a study of convention. See Skepticism and Objectivity.
It is self-undermining. The doctrine is stated in concepts — "universal", "name", "convention", "arbitrary" — each of which must genuinely apply to its instances for the sentence to say anything. See Fallacy of the stolen concept.

The third alternative

Nominalism and realism are usually presented as the only options: either universals are real entities existing apart from or within particulars, or they are only words. Objectivism holds both to be halves of one false alternative, the same pairing as intrinsicism and subjectivism.

Its own position is that concepts are neither found ready-made nor invented at will. They are formed — by a specific, identifiable mental process, performed on real similarities among real things — and they are objective because the process is one that can be checked and repeated. See Objectivity, Concept and Concept formation.

On this view a definition is neither an arbitrary stipulation nor a report of an eternal essence, but the identification of the fundamental characteristic of the units within a given context of knowledge. See Rule of Fundamentality.

Common misunderstandings

That rejecting nominalism means accepting Platonic forms. Objectivism rejects both. Universals exist neither in a separate dimension nor as metaphysical essences inside particulars; they exist as the products of a mind performing abstraction on real facts.
That concepts being man-made makes them arbitrary. A bridge is man-made and does not therefore hold arbitrary weight. What determines whether a concept is valid is whether the process that formed it observed the requirements of concept-formation.
That "words are conventional" implies nominalism. Which sounds are used to designate a concept is genuinely conventional. What the concept subsumes is not. See Concept.
That it only concerns metaphysics. Applied to values, nominalism becomes the position that moral terms name no real property and merely express approval — which arrives at subjectivism in ethics.

Discussion

References

Rand, Ayn. Introduction to Objectivist Epistemology. 2nd edition, Meridian, 1990 — the introduction states the problem of universals and the rejected answers.
Peikoff, Leonard. "The Analytic-Synthetic Dichotomy", in the same volume.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapters 3 and 4.