Wikipedia explains the analytic-synthetic dichotomy as follows:
The analytic–synthetic distinction (also called the analytic–synthetic dichotomy) is a conceptual distinction, used primarily in philosophy to distinguish propositions into two types: analytic propositions and synthetic propositions. Analytic propositions are true by virtue of their meaning, while synthetic propositions are true by how their meaning relates to the world. However, philosophers have used the terms in very different ways. Furthermore, philosophers have debated whether there is a legitimate distinction.The first problem we confront in Peikoff's essay is the following claim:
The tenets underlying [the analytic-synthetic dichotomy] permeate our intellectual atmosphere like the germs of an epistemological black plague waiting to infect and cut down any idea that claims the support of conclusive logical argumentation, a plague that spreads subjectivism and conceptual devastation in its wake.All of these claims are, of course, grossly implausible. Most people neither care about nor even comprehend the analytic-synthetic dichotomy. Nor is it true that "virtually every influential contemporary philosopher" accepts the theory "in some form." I'm no expert in contemporary philosopher, but even I know that Willard Quine's essay "Two Dogmas of Empiricism," which attacks the analytic-synthetic dichotomy, is one of the most influential philosophy papers of the 20th century. Quine's essay was published 1951, more than fifteen years before Peikoff's attack on the analytic-synthetic dichotomy appeared in The Objectivist. Peikoff's assertion that "virtually every influential contemporary philosopher" accepts the theory suggests that Peikoff either (1) had never read Quine's essay; or (2) that he is just making things up for rhetorical effect.
This plague is ... accepted, in some form, by virtually every influential contemporary philosopher -- pragmatist, logical positivist, analyst and existentialist alike.... The theory ... penetrates every corner of our culture, reaching, directly or indirectly, into every human life, issue and concern. It's carriers are many, its forms subtly diverse, its basic causes complex and hidden -- and its early symptoms prosaic and seemingly benign. But it is deadly. [IOTE, 89]
Why does Peikoff attack the analytic-synthetic dichotomy? There are several reasons, but the main one stems from Rand's fondness for definitions and tautologies, i.e., for "analytic" truths. The analytic-synthetic dichotomy exposes the triviality and empirical vacuity of analytic statements; therefore, Peikoff had to be sent, pen in hand, to vanquish it. It is important not to get sidetracked by Peikoff's bold assertions about the horrors of the ASD. It's all nonsense and window dressing. What's really at issue is the Objectivist view of the relation between logic and fact and Rand's naive belief that tautologies such as existence exists and A is A constitute significant insights into the nature of reality.