The existence of this supposed distinction has been a hot topic in professional analytic philosophy for the last 50 years. Here's a sketch of what the distinction is, and of one famous argument against its existence.

Before Quine, it was thought that there are two kinds of statements:

analytic statements, which are true or false solely in virtue of the meanings of the words in them (i.e., true or false by definition), and
synthetic statements, which are true or false depending on how things are in the world.

For example:
analytic statements:

synthetic statements: (whether they are true or false depends on something about the world, something other than the meanings of the words)

• There were exactly two bottlecaps on nate's kitchen table at midnight, 8/20/01.
• The sky is blue today.
• The oldest E2 user is 107 years old.

To see if one of the synthetic statements is true, we have to check in the world (make observations). That is, the truth value of a synthetic statement depends on how things are in the world. But the analytic statements are different: if we understand the words involved, we thereby know whether the statement is true -- we don't have to check in the world. Truths of logic and basic mathematics are classic examples of analytic truths, whose truth value does not depend on how things are in the world.

Quine argued that the supposed distinction between these types of statements is nonexistent.

His argument began from the premise of holism about beliefs -- all of our beliefs (from facts of arithmetic to empirical observations about the room we're in right now) are inter-related in a kind of web. It's been traditional to think that truths of arithmetic and logic are "fixed" points in this web -- that no observation could dissuade us from our belief in them. But, Quine says, this is not so. If we were willing to accept enough weird changes elsewhere in the web, the truths of arithmetic would not be immovable, fixed points. In his famous phrasing: "Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system."

Notice that he's not recommending we embark on goofball belief-changing projects! Instead, he's pointing out that (contrary to what logical positivists thought) it's not strictly logical considerations that determine which beliefs are "fixed", but rather pragmatic considerations.

The initial paper in which he argues this is "Two Dogmas of Empiricism", 1951? 3? (In the paper, he describes and rejects two beliefs held by logical positivists -- the empiricists of the title. The existence of the analytic-synthetic distinction is one of these two beliefs --"dogmas" -- he rejects. The other dogma is reductionism, which he described as "the reduction of all meaningful statements to statements about immediate experiences"; i.e., he was rejecting the positivist doctrine of verificationism.)

Not everyone in analytic philosophy is convinced by Quine's arguments; the alleged distinction is still a matter of debate. But the very thought that it could be debated was pretty revolutionary.

(Analytic and synthetic statements have been thought by some -- notably, the logical positivists -- to correspond to two kinds of knowledge we can have about the world: a priori knowledge, and a posteriori knowledge. This correspondence has been challenged by Immanuel Kant, and more recently Saul Kripke and Ruth Barcan Marcus. It is not my subject here. See synthetic a priori?)