First, let's call things by their right name to avoid confusion: der, die or das can mean different things depending on the context: They can be definite articles, relative pronouns, or I could take the question to mean "when are things masculine, feminine or neuter"?. As I understand your question, what you're asking really is "when should I use a definite article instead of an indefinite article"?
Given that both German and English have the same features when it comes to definite and indefinite articles, I can't think of any general case where you would use one in English and the other in German. However, there are cases where the usage of each type of article differs.
Professions: No article is used in German when stating someone's profession.
He's a doctor
Er ist Artz
Nationalities: Whereas the English sentence works with the article, the equivalent German sentence doesn't.
I'm [an] American
Ich bin Amerikaner/in
A gram of safron costs $8
(Saffron costs $8 per gram)
Safran kostet 8 € das Gramm
Names: For certain names it is common for some Germans to use a definite article. This doesn't happen in English (that I'm aware of).
Ich habe gerade mit dem Maxi telefoniert!
Hair: In English, hair is almost exclusively a non-countable noun; in German it can be used in the plural (usually to the amusement of English-speakers)
My hair was standing on end!
Die Haare standen mir zu Berge!
Places/locations like work and church are not written with articles in English; in German, they are.
I'm going to church
Ich gehe in die Kirche
Ich besuche den Gottesdienst
Fixed expressions Canoo has a list of fixed of expressions which can deviate from the normal rules for articles (the page is in German, though). It also a lot of examples of both definite and indefinite article usage.