Metaphysics consists of knowledge apprehended by pure reason. By definition, metaphysics studies what is beyond experience. The Greek root meta means "beyond," so "metaphysics" literally means "beyond physics." Like mathematics, metaphysics is an a priori body of knowledge.
The distinction between a priori and a posteriori ways of thinking is that the former are drawn from pure reason and the latter are drawn from experience. Kant goes on to draw a second, even more important, distinction between analytic and synthetic judgments.
The predicate of an analytic judgment is contained in the concept of the subject: the predicate, then, is simply an analysis of the subject concept. "All bachelors are unmarried" is analytic: being unmarried is a part of the concept of "bachelor," so saying that all bachelors are unmarried doesn't add anything to our concept of "bachelor"; it just clarifies the definition.
The predicate of a synthetic judgment, on the other hand, adds something new to the concept of the subject: it synthesizes two different cognitions. "All swans are white" is synthetic: we can know what a swan is without necessarily knowing that it's white, so learning that swans are white is an additional cognition that we can attach to our concept of "swan."
All analytic judgments are a priori, since they consist simply in the analysis of concepts and do not appeal to experience. Synthetic judgments, on the other hand, can be either a priori or a posteriori. Kant classifies synthetic judgments into three types: judgments from experience, mathematical judgments, and metaphysical judgments.
Judgments from experience are synthetic a posteriori since they are pieced together (synthetic) from the objects of experience (a posteriori).
Mathematics consists of synthetic a priori judgments. The concept of "7 + 5," Kant argues, contains the union of those two numbers in a single number, but the concept itself does not contain the number 12. We must make a leap of intuition in order to determine that twelve is indeed the number that results from the union of seven and 5. The same is true of geometry: the concept of the shortest distance between two points is not contained within the concept of a straight line. The temptation to think of math as analytic comes from the fact that the truths of mathematics are necessary: we cannot reasonably deny that 7 + 5 = 12. The fact of the matter is that mathematical cognitions require intuitive leaps that are synthetic in nature.
Metaphysics also consists of synthetic a priori judgments. It may seem that metaphysics consists largely of analytic judgments, since the only thing metaphysicians agree upon are the various definitions that are analytic in nature. However, metaphysics consists of synthetic judgments that are built upon these analytic definitions, much like mathematics consists of synthetic judgments built upon analytic axiomatic truths.
The need to ask whether metaphysics is even possible arises because there is little agreement over the synthetic judgments that ought to constitute it as a body of knowledge. Kant's proposed method is to start with the assumption that synthetic a priori judgments are possible, since they constitute both mathematics and pure natural science. He will investigate how synthetic a priori knowledge is possible in these fields in the hopes of discovering also how such knowledge might become a reliable source for metaphysics. He proposes to examine first mathematics, then pure natural science, and then ask how metaphysics is possible in general and as a science.
The distinction between a priori and a posteriori shows the two possible sources of knowledge: the intellect and experience. If we can know something independently of experience, it is a priori, and if we know something through experience, it is a posteriori. Math is a paradigmatic example of a priori knowledge: I can figure out that 7 + 5 = 12 in my head, and nothing I find in experience can possibly contradict that knowledge. The statement "all bachelors are unmarried" is also a priori even though it refers to bachelors, which, unlike numbers, can be found in the world outside our heads. The reason is that "all bachelors are unmarried" is a definition of a bachelor rather than a statement based on experience. The statement, "all bachelors are lonely," on the other hand, is a posteriori, since loneliness is not a part of the concept of "bachelor." That statement is drawn from the speaker's experience with bachelors or from what other people have told him about bachelors.
While the a priori/a posteriori distinction is epistemological, distinguishing between sources of knowledge, the analytic/synthetic distinction deals with the logical structure of the judgments themselves. "All bachelors are unmarried" is analytic because the concept of bachelor is "unmarried man": this statement merely clarifies a part of the concept of "bachelor." A good test for whether a statement is analytic is to ask whether people could understand the subject concept if they did not know that the predicate were true of it. For instance, if I did not know that all bachelors are unmarried, I couldn't properly be said to understand what a bachelor is. On the other hand, "all swans are white" is synthetic because, even though we may generally think of white animals when we think of swans, I could be said to understand what a swan is without knowing that it is white.
Kant was the first person to draw the analytic/synthetic distinction explicitly. Until Kant, this distinction had generally been lumped together with the a priori/a posteriori distinction. Hume and others had considered the propositions of mathematics to be analytic. In proposing that there are synthetic a priori judgments, Kant suggests, contrary to conventional wisdom, that propositions like "7 + 5 = 12" are in fact synthetic. His argument is essentially that the concept of "7 + 5" is the union of the concepts of "7," "5," and addition. None of those three concepts in themselves contain the concept of "12"; it is a new concept that arises from the synthesis of the three subject concepts.
There are further arguments in Kant's favor. If concept of "12" were part of the concept of "7 + 5," then so would the concepts of "9 + 3" and "16 - 4," and an infinitude of other concepts. How could the concept of "7 + 5" possibly contain all these other concepts? Also, it seems ridiculous to suggest that the concept of "154,938" is a part of the concept of "52,624 + 102,314": we could understand the concept of that sum without necessarily knowing what the two numbers add to.
While analytic judgments consist simply of analyzing the subject of a proposition, synthetic judgments add something new to it. They effectively connect two independent pieces of knowledge to each other. Kant explores the nature of this connection. With synthetic a posteriori knowledge, the connection is made through experience. If I see a lot of white swans, I come to associate the concept of white with the concept of swan through experience. With synthetic a priori knowledge, the answer is more complicated. How do I learn to link the concept of "7 + 5" with the concept of "12"? In the sections that follow, Kant sets about trying to answer that question. His hope is that if he can explain how we can connect concepts in pure mathematics and pure natural science, he will also be able to explain how we can connect concepts in metaphysics.
The analytic/synthetic distinction is one of Kant's most significant contributions to philosophy. Like any significant contribution, it has been the subject of heated controversy ever since. One of the difficulties lies in saying precisely what the concept of "bachelor" or "swan" consists of. At what point can I no longer properly be said to understand what a thing is? If all humans are animals and all humans have noses, why is being an animal a part of the concept of being human and having a nose not? The idea that words have "concepts" attached to them goes all the way back to Aristotle's essences.