The synthetic/analytic distinction is fundamentally a linguistic or semantic distinction, but it may have epistemological or metaphysical consequences.
In the way it is most commonly used today, an analytic proposition is understood to be one that is true in virtue of the meanings of the terms it contains. The idea is that the meaning of 'puppy', 'young' and 'dog' are sufficient to ensure the truth of "puppies are young dogs", because that is how the meanings of the words are related. The original meaning of 'analytic' was defined by Kant in terms of a containment relation, but that turned out to be too narrow and it is no longer used.
The potential significance of the distinction is that it may be deployed to explain why some propositions are considered to be a priori knowable, and/or why some propositions are considered to be necessarily true. A priori/a posteriori is an epistemological distinction: it is concerned with whether we can know a proposition to be true independently of our empirical experience of the world, at least once we have the minimum experience needed to grasp what the proposition means. Necessity/contingency is a metaphysical distinction: it is concerned with whether a proposition must be true or whether it could have been otherwise.
As far as I know, it is generally accepted that all analytic propositions are a priori knowable, since if you know the meanings of the words, and the meanings of the words guarantee the truth of the proposition, then you are in a position to know the truth of the proposition. The converse, i.e. that all a priori propositions are analytic, is highly controversial. Kant believed there were synthetic a priori propositions. The logical positivists claimed that there were not. Modern philosophers are divided on the issue (see, for example, this question. What are the more complex/interesting examples of synthetic a priori statements?).
When it comes to the necessary/contingent distinction, Kripke's views on necessity are highly influential, and on his position there is a sharp difference between this and the a priori/a posteriori one. So much so, that according to Kripke there are propositions that are a posteriori and necessary, and there are propositions that are a priori and contingent. Kripke proposes to define 'analytic' as a proposition that is both necessary and a priori.
As to what is at stake with the analytic/synthetic distinction, for the logical positivists, it was a way to explain why some propositions are a priori and/or necessary. The logical positivists wished to dismiss as nonsense all kinds of metaphysical claims that they considered to be unscientific. By denying the synthetic a priori they claimed that a priori knowledge is knowledge only of analytic propositions, which is uncontroversial. By denying synthetic necessities, they were claiming that there were no essential properties, i.e. that all necessity is de dicto only, and that necessary propositions are in turn just analytic, and hence uncontroversial. If these claims worked, they would simplify, and indeed eliminate, a great deal of what philosophers talk about. But logical positivism is pretty much dead and buried these days, so clearly the majority view is that these claims don't hold up.
The other point to note about the analytic/synthetic distinction is that many philosophers don't accept it at all. Quine attacked it in a series of papers, including "Truth by Convention", "Two Dogmas of Empiricism" and "Carnap and Logical Truth". For Quine, the important difference between propositions (he prefers to speak of sentences) is how exposed vs. how far protected they are from empirical revision. But none are entirely immune to revision in the light of experience. On this view, there is no binary analytic/synthetic distinction, nor a binary a priori/a posteriori distinction, just a spectrum. So there are no consequences for scientific knowledge either way.
There is a good account of the relationship between the analytic/synthetic, a priori/a posteriori, and necessary/contingent distinctions in A.C. Grayling's textbook, "An Introduction to Philosophical Logic", chapter 3. Kripke's main text is "Naming and Necessity".
