First of all, "der", "das", and "die" are definite nominative articles, and they all translate as "the". German, as opposed to English, uses grammatical gender. Those articles are randomly attributed to nouns and change depending on the case (nominative, accusative, genitive, dative). The correct translation for "The apple I'm eating is tasty." is: "Der Apfel, den ich esse ist lecker." In this sentence, "Der Apfel" is nominative and "den" is the accusative form of "der". In sentences like "the apple that I eat" (der Apfel den ich esse), "the car that I see" (das Auto das ich sehe), and "the lamp that I buy" (die Lampe die ich kaufe), the English word "that" is replaced by a corresponding article. This is different in sentences like "I know that this is right." ("Ich weiß, dass das richtig ist."). Here "that" translates as "dass" since it's not referring to an object that is eaten, seen, bought, etc. The fact that you can omit "that" in some sentences in English does not change the German sentence.
