A priori vs. empirical knowledge. Kant takes for granted the idea of a priori knowledge, but he provides what amounts to an explanation of the idea in the third paragraph of §1 (p. 9). Although the natural opposite term is “a posteriori,” Kant will more often oppose it to empirical knowledge, so you might think of the a priori as the non-empirical. But you should avoid thinking of it as innate knowledge in the sense discussed by Descartes and Locke. Kant rejected that idea of its source in his comments on Plato and Crusius in the letter to Herz (pp. 119f). What Kant takes to be its source will be a central topic of the Prolegomena, and he will later add to the “pure reason” and “pure understanding” he mentions in §1 (and will explain later), an idea of “pure intuition.” The latter is related to empirical experience, so you also should not think of the a priori as completely disconnected from the empirical. Kant’s term “pure” may help here since he tends to use it to speak of things that do not involve particular empirical content, and that’s the key feature of a priori knowledge.

Analytic vs. synthetic judgment. Kant explains this distinction in the first two paragraphs of §2 (pp. 9f) and says more about it thereafter. I mention it here only to note its difference from the distinction below, which you already encountered in the preface (p. 8).

Analytic vs. synthetic method. While the distinction between analytic and synthetic judgment was introduced by Kant, he has a traditional distinction in mind when he distinguishes an analytic from a synthetic method. (He discusses the difference between the two distinctions in a footnote to §5, p. 18 n. 6.) The second analytic/synthetic distinction derives from labels used in ancient mathematics for the two parts of an approach to solving mathematical problems. When faced with the problem of providing a geometric construction, a good approach is to imagine you already have the completed construction and to distinguish various simpler constructions that would be required (and enough) to achieve it. You can then apply the same approach to these constructions in hopes of eventually reaching ones you know how to produce. This is the first part of the “method of analysis and synthesis,” and Kant thinks it might be better labeled “regressive” since it moves backward from the desired result. Once you have moved backward far enough to know how to start, you can actually construct the figure; and this construction is the second part of method of analysis and synthesis, which Kant labels “progressive.” (Descartes has the method of analysis and synthesis in mind when he speaks in the Discourse of breaking a problem into its smallest parts, and his approach to geometry via connections with algebra is often called “analytic geometry” to distinguish it from the proof of theorems from geometric axioms, which generally employs constructions and is called “synthetic geometry.”)