Knowing when to use the proper article before a noun in Danish("en" and "et") has to be one of the biggest challenges of learning Danish. Native Danish speakers can easily differentiate between foreigners and immigrants, and those who skipped elementary school or their Danish language classes, by their incorrect or correct usage of the articles "en" and "et", as there is hardly any way of logically deducting when to use which one of the two. When to use "et" or "en" is something you simply have to know by heart, somewhat similar to way that you have to know the gender of a noun in French and German.
Further explanation: All nouns in Danish get the article (the) "embedded" at the end(suffix) of the noun when it is in its definitive form.
So the definite sentence "the man" in English, would translate into "manden" in Danish. The indefinite sentence "a man" translates into "en mand".
However, there are exceptions to this rule: some nouns end on "et" instead of "en" in their definite and indefinite forms.
An example of this would be the (definite) sentence "the house", which translates into "huset" in Danish. The indefinite sentence "a house" translates into "et hus" in Danish. An easy way to figure out what when to use "en" or "et" is to simply look at the last letter of the noun; if it ends on a "t", like "bordet"(the table), you would say "et bord"(a table). And vice versa, if the noun ends on a "n", like "spanden"(the bucket), the appropriate article would be "en spand"(a bucket).
75% or more of Danish nouns follow the before mentioned rule.
Another hurdle you have to get over with Danish, is the muddled nature of pronunciation: When speaking Danish, often the ending of a sentence or word is "swallowed" by the speaker, making it hard for a non-native speaker to make out the word ending and the usage of "en" and "et".
I hope this helps!
This is something that many people struggle with, but is very useful to learn as soon as possible.
As ioad wrote, the article is the ending of the noun.
a woman = EN kvinde the woman = kvindEN
a man = EN mand the man = mandEN
an apple = ET æble the apple = æblET
The use of the article 'en/et' in danish does not have the same rules as in English where you in general use 'a', but uses 'an' if the noun starts with a vowel-sounding letter. In danish I am quite sure there isn't these rules, it is simply something you have to know. Same as in German where you just have to know if the noun is male, female or neutral.
I hope it helps