The indefinite article ("a" or "an") is either a "pick from a collection" word or a new information marker in English. The definite article ("the") is either a specifier or refers to something that has already been introduced (old information, if you will). Think of "the" in most cases as a sort of weakened version of "that", and it may help. It is a subtle distinction, and one which may quickly become intuitive to a native speaker, but not one that a native speaker can always explain when asked. (Europeans generally have less trouble, but definite and indefinite articles are a European areal feature.) It doesn't help at all that the rules are often deliberately broken in literature to force the reader into a scene (usually by using a definite article for a first introduction, as if you'd known about it all along).

A cat appeared on my back porch this morning. (New cat, no history at this point.) The cat then proceeded to shred the upholstery on every bit of furniture it could find. (We've already met the cat; using "a" in this sentence would introduce another cat.)

Of course John speaks a bit of Portuguese; he married a Portuguese woman. (We may have known John's wife for many years, but in this case she is merely one of a collection of Portuguese women. If we had used "the" in this sentence, it would imply that there is only one Portuguese woman in the world, or at least in the area, and that John married her.)