Another area that German is more complex than English, is cases. In English, we don't have any! However, there are 4 cases in the German language! Nominative, accusative, dative and genitive.
When the noun is the subject of the sentence, the nominative case is used. What changes for each case are the articles. Nominative is the easiest to remember, as the articles used are the same as the ones taught above. ie "der", "die", "das", "ein", "eine", "ein". Obviously, which article is used depends on the gender of the noun.
The accusative case is a little more complicated. It is used when for the object of the sentence. In this case, "der" turns into "den" and "ein" into "einen".
The dative case is used to the indirect subject of the sentence. In the dative case, "der" turns into "dem", "die" turns into "der" and "das" turns into "dem". Also, "ein" turns into "einem", "eine" turns into "einer" and "ein" turns into "einem".
The final case (yes that's all) is called the genitive. It is used when you refer to posession. Check the table below for all the correct forms when a certain case is applied.