To get better at hearing a foreign language, you just have to hear it a lot. That's why my Spanish teacher talks in Spanish all forty-five minutes of every class unless you specifically ask her- in Spanish- to repeat the directions in English. I used to think it was annoying, but it actually helps a lot. The more you hear it, the more your mind adjusts to it and recognizes it. After a long time, you can easily think in that language just like you can in English.
masc = masculine fem = feminine pl = plural
nominative masc: ein/der accusative masc: einen/den dative masc: einem/dem genitive masc: eines/des nominative neut: ein/das accusative neut: ein/das dative neut: einem/dem genitive neut: eines/des nominative fem: eine/die accusative fem: eine/die dative fem: einer/der genitive fem: einer/der nominative pl: eine/die accusative pl: eine/die dative pl: einen/den genitive pl: einer/der
Go here for a chart with all this information- it's a lot easier to look at that than to memorize all this: http://en.wikipedia.org/wiki/German_articles
The article changes depending on the grammatical gender, number and case of the noun:
In the correct translation, it's den Apfel because the noun is accusative case, singular and masculine, and so is the article.
die on the other hand would be the right article to use for…
… the nominative (roughly equal to subject case) singular for feminine nouns:
„Die Birne schmeckt gut.“ – „The pear tastes good.“
…the accusative (roughly equal to direct object case) singular for feminine nouns
„Er hat die Birne.“ – „He has the pear.“
… nominative and accusative case for plural nouns, in all three genders:
„Er hat die Äpfel.“ – „He has the apples.“
„Die Birnen schmecken gut.“ – „The pears taste good.“, etc.
der is (among others) the nominative case (“subject case” for now) definite article for a masculine singular noun.
„Der Apfel ist rot.“ – “The apple is red.”
den is (among others) the accusative case (“direct object case”) definite article for it.
„Er hat den Apfel.“ – “He has the apple.”
If you need a short introduction to the German cases, I suggest you to read this article
The word has the same meaning in English. The form of the word changes depending on its gender and where it appears in the sentence.
Genders: Der, masculine. Die, feminine. Das, neutral.
Cases: Den, accusative. Action done to the noun.
Das Auto, Der Stuhl. Das Auto fuhr über den Stuhl. THE car ran over THE chair.
Der Stuhl fuhr über den Auto. THE chair ran over THE car.
der Apfel is singular, the apple, die Äpfel is plural, the apples...
der and die are used when the noun is the subject of a sentence: Der Apfel ist rund. Die Äpfel sind rund.
den and die are used when the noun is the object of a sentence: Er isst den Apfel. Er isst die Äpfel.
No. den has quite a few meanings, but the most important ones at the beginning of the skill tree are probably
the dative (roughly equal to indirect object case if you haven't been introduced to dative yet) plural of the definite articles der / die / das:
„Er schickt den Kindern einen Brief.“ – „He sends a letter to the children.“
the accusative (roughly equal to direct object case) singular of the masculine definite article der:
„Er schickt seinen Kindern den Brief.“ – „He sends the letter to his children.“
den is never an indefinite article and thus never translates to a or an.