More precisely: En for common gender words such as e.g. mand, kvinde, but also hoste (cough) and in fact the majority of Danish words. And et for neuter gender words such as e.g. vand but also barn (child) and many words for animals. Unlike in English, gender is a purely grammatical property of a word.
This goes back to the Proto-Indoeuropean language(s). Proto-Indoeuropean originally also had two genders, the common gender and the neuter gender. The common gender later split into masculine and feminine. E.g. German and most Norwegian dialects still have three genders just like Old English and ancient Latin and Greek. In the Romance languages, the neuter gender was essentially merged into the masculine gender, leaving only masculine and feminine. In Germanic languages the trend is to re-merge masculine and feminine into a revived common gender, as in Dutch and the Scandinavian languages.
Right before gender was lost in English (about 13th century), the was the definite article and demonstrative for common gender, and that was the one for neuter gender. If this distinction had not been lost, Modern English would be a bit like this: "The man and the woman went into that house to see what that child was doing with the hound." You can still sort of feel the connection if you go through German and Dutch:
- English the - German and old-fashioned Dutch den (as in Den Haag) - Danish en.
- English that - Dutch het, dat - Danish et.
(Slight clarification: Gender is purely grammatical in other languages, too---and many non-Indo-European languages have quite a bit more than 3. [Some African languages have over a dozen.] Other languages, including some IE families, divide by 'animate/inanimate' rather than 'masculine/feminine'. The word gender derives from a Latin word meaning simply 'category'.)
Æ, æ is the Danish and Norwegian version of the umlaut (a German word meaning something like changed sound) that is written Ä, ä in Swedish and German. Both variants started as ae.
The combinations ae, oe and ue used to occur frequently in Germanic languages because there were grammatical rules (e.g. for forming the plural) that added an e after a, o ,u in certain situations. Originally this was pronounced simply as a sequence of two vowels, but later people just changed the pronunciation of the first so it sounded more like the second, and dropped the second vowel. So it made sense to consider the two letters as a unit in writing as well. The Danish variant is the result of simply writing a and e close together. The Swedish/German variant started by putting a tiny e on top of the other vowel. This was later replaced by two dots for simplicity.
Nowadays the umlaut sounds and letters also occur in words that are not the product of some grammatical rule. And other languages such as Finnish, Hungarian and Turkish have adopted some of the letters even though they never had umlauts in the strict sense of the word.
I have no idea how the umlauts are pronounced in Danish, because Danish phonetics is just plain weird (see this satire on the problem) and I am not even turning on the sound for my Danish lessons. I think the phonetic differences between Danish and the other Scandinavian languages may be similar to those between French and the other Romance languages. I just want to be able to read all Scandinavian languages. Understanding Danish will take some getting used to, and I guess learning to speak it won't really make much sense for me as Danes should have little trouble understanding a Swedish or Norwegian pronunciation and I am unlikely to travel to Scandinavia anyway.
As a German native speaker I am only really qualified to explain the phonetics of the German umlauts, but I think in Swedish and Norwegian it's very similar.
German for bar is Bar, and German for bear is Bär (spelled Baer when no umlauts are available). The differences in pronunciation for these words between German and English aren't greater than those between variants of English such as British and American. As you can see, the es in the English words bear and bare have the same effect on the pronunciation of the a as the e or two dots in German. Only in the more regular German orthography, the position of the e is standardised, whereas in English it retains its original position from the time when all vowels were pronounced separately.
The difference between German o and ö is like that between the English words born and burn. (In Danish and Norwegian, ö is written as ø.)
The only really hard one for English speakers is ü, which doesn't exist in the Scandinavian languages. (U is easy, it's like the vowel in food. But ü is like the sound you have to go through when moving from i (as in tin or either vowel in easy) to this u. You can think of ewww as the sequence of German vowels iüu.)
All Scandinavian alphabets also have the special letter å, the result of combining a and o to describe a sound that is somewhere in between. The origin of this letter, however, is the sequence aa, and that's how you have to transcribe it if no å is available. As the letter å is a Swedish innovation that only reached Danish and Norwegian in the mid-20th century, I think one can still encounter aa a lot in the latter two languages.
"En" (and "et") mean "a" or "an". Which one you used depends on the gender of the word, which is usually random, with a few rules about suffixes and compound nouns. For more information, see this discussion: https://www.duolingo.com/comment/4288076
"Er" is the present tense form of the verb "at være" ("to be"). This is always the present tense form and does not change depending on the subject like it does in some other languages.
But en and en do. In Danish, as in most languages that have a singular indefinite article (including English), it is derived from the numeral for 1. As in many languages (but unlike in English), it is not (yet) differentiated from that numeral. An or even a (before a consonant) is what comes out when you pronounce one very carelessly.