Translation:The cool elf makes a cool Christmas present for the cool boy.
This sentence includes two "fede" and one "fed". If you look at the noun groups at hand, you will see a pattern:
1) Den fede nisse
2) En fed julegave
3) Den fede dreng
The first and the third start with den, the second starts with en. As you might already know, "den/det/de" mean "the" (Den for nouns ending in en, det for nouns ending in et and de for all plural nouns, no matter what their ending is): Whereas "en/et" mean "a/an".
"The" is called a definite article, since it points to something definite, i.e. something specific (The apple). Whereas "A (or an)" is called an indefinite article because it points out to any one of the (noun) (An apple).
The two rules governing these articles in Danish are:
1) If an indefinite article is used; then the formula is:
en+ adjective according to en + noun (en fed gave)
et+ adjective accroding to et + noun (et fedt aeble)
2) If a definite article is used; then the formula is:
den + adjective in plural form + noun (den fede gave)
det + adjective in plural form + noun (det fede aeble)
de + adjective in plural form + noun (de fede gaver / de fede aebler)
NB: "The" is not the only definite article, there are others like my/your/her or that/this/those/these etc.
Good answer, except for the definition of word class article which I would like to adjust a bit, hope it is ok...
While 'the' in English as a definite article, 'den', 'det' and 'de' are actually not in the Scandinavian languages; Our definite articles are the -en/-et/-ene endings of the nouns (and adjectives, you could argue). The words 'denne'/'den', 'dette'/det' and 'disse'/'de' are demonstrative pronouns like 'this'/'these' and 'that'/'those' in English...ALTHOUGH... I completely agree that the semantic meaning is closer to 'the' in English...unless the pronoun is really stressed/emphasised by the speaker, in which case I'd say it turns into being more like English this/that... (The word 'den' plays a different role in a general expression like 'Har du set den glade landmand?' vs. 'Den landmann [pointing at the person] er ret glad!' The latter focuses on the individual as opposed to farmers in general.)
I've also seen, however, that some linguists have referred to these demonstrative pronouns as "adjective articles" a couple of times, which might be a good description, and which might be bridging the two different semantic translations to English... :-)
Den danske Ordbog's entry on the adjective "fed" has meaning "fat" including application to people as definition 1. The slang one meaning "cool" is way down at definition 7. https://ordnet.dk/ddo/ordbog?select=fed,4&query=Fed