In most of the cases, it is the same as in English: I got a cat: Ich habe eine Katze. (indef.) I got cats: Ich habe Katzen. (indef.) The cat is black: Die Katze ist schwarz. (def.)

No article at all is just the plural form of the indefinite article, the so-called zero article. When something is mentioned the first time in a text, usually it has the indefinite article. In all subsequent occurences the definite article is used for it. But this does not apply to objects that are unique and/or can be assumed to be generally known ("die Menschheit", "der Teufel", "das Oktoberfest" ) or that are specified in a distinct way. -> "Mir kam ein Mann entgegen. Der Mann hatte einen Hund bei sich. Der Hund hatte ein braunes Fell und Schlappohren." Here "Mann", "Hund", "Fell", and "Schlappohren" have indefinite articles upon first mention and definite ones on subsequent mention. "Schlappohren" is plural and therefore has the zero article. -> "Der Mann, der mir entgegenkam, hatte einen Hund bei sich. Das Fell des Hundes war braun und er hatte Schlappohren." Here "Mann" and "Fell" are well specified, "Mann" by a subordinate clause and "Fell" by a genitive attribute. Therefore they have the distinct article right from the beginning.