The 'narrator' pronounces l(e)sen with a hard e (e in question in parenthesis). I thought e was pronounced more like 'eh' and only i was pronounced like 'eeee'. Is the change because there is another vowel opposite to the letter next to it (as English vowels can change like that)? Or is this just a case I have to memorize?
"The boys are reading a newspaper" would be "Die Jungen lesen eine Zeitung". The gender of "Zeitung" is feminine, thus, as sparroe said, it is "die Zeitung". "Die" is a definite article just as "the" is, where as "eine" (and other forms thereof, depending on case or any prepositions that may affect case) is an indefinite article, just like a/an.
I'll start with nominative and accusative, as they are the easiest (in my opinion).
Nominative case: The subject of the sentence takes on this case. The subject is the person doing the action in the sentence. In English, we determine the subject through the order of the words in the sentence.
"The girl eats the apple" and "the apple eats the girl" mean two different things. In the first sentence, the girl is the subject, and the apple is in the second sentence. They will take on nominative cases, and thus these articles:
Der - masculine Die - feminine Das - neuter
In those sentences, the direct object, which takes on the accusative case, is the thing the action is being done to. The first sentence's object is the apple, and the second sentence's object is the girl. In the accusative case, these are the articles used:
Den - masculine Die - feminine Das - neuter
In this case, as apple is masculine, you can determine whether it is the subject or the direct object by its article, and thus sentence order doesn't matter.
"Das Mädchen isst den Apfel" and "Den Apfel isst das Mädchen" both mean "The girl eats the apple."
"Der Apfel isst das Mädchen" and "Das Mädchen isst der Apfel" both mean "The apple eats the girl."
NOTE: There is a huge and very common exception to the accusative case. When using the verb 'to be' (sein, sind, bin, ist, seid), both sides of the sentence take the NOMINATIVE case. (Also, I know my examples don't make sense, but they get the point across.)
"The girl is the apple" "Das Mädchen ist der Apfel." "Der Apfel ist das Mädchen."
Hope this helps any who were confused! Worry about dative later - it's a little harder to deal with in my opinion, but has to do with the indirect object of a sentence.