"Den" refers to?
"Ich sehe den Apfel."
Do you see this apple? Do you see that apple? Do you see the apple?
If it were just "the apple" I can say: "Ich sehe Apfel." No?
It is "the apple" in Accusative case. If you say "Ich sehe Apfel" it would bee something like: "I see apple(s)", in general. You have some charts with the cases http://de.wikipedia.org/wiki/Artikel_%28Wortart%29
It depends on the case end gender. They are all translated with "the" in English, but in German der (m), die (f), das (n) and die (pl) (in Nominative case) change as follows: Akk: den (m), die (f), das (n) and die (pl) Dat: dem (m), der (f), dem (n) and den (pl) Gen: des (m), der (f), des (n) and der (pl)
where m=masculine f=feminine n= neutrum pl=plural
I learned this the hard way from a different language, but any simple sentence can have 3 main parts: The person/thing doing the doing. The person/thing the doing is being done to. The place (to, from, with, at, etc) the thing is being done. Der Apfel is the apple doing the doing. Den Apfel is the apple is being done to. Dem Apfel is the "directional" (as I had learned to call it.)
So "Der Mann (the doer) sieht (the deed) den Apfel (what the deed is being done to) auf dem Tisch ("on the table", where the deed is being done.)"
In German, they call those cases. They are handy because then you can start changing up word order to emphasize different parts of the sentence. (Den Apfel ich esse = the apple is being eaten by me.)
I'm still fuzzy myself on the different ways to say "this" and "that" in addition to "the" but I suspect they might all be somewhat interchangeable.
Hope that helps. Mostly, until you get to the lessons on cases, it should be just enough to know that Apfel is masculine.