Yes, they both translate to 'a/an(/one)' in English.
et is the indefinite article for neuter nouns.
en is the indefinite article for masculine nouns.
ei is the indefinite article for feminine nouns.
PS: All feminine nouns can also be declined as if they were masculine, using the masculine declination pattern, so you can choose to use en for feminine nouns as well.
I had the same problem too. You should bite the ends of words, I.e. make the ends of words sort of melt together, e.g. piger = "piie" or something like that. Give it a shot, it worked for me! ;)
Et barn is a child, literally. The Et tells you that it is a child, and that child is a neutral noun, i.e. neither male nor female. If you wanted to speak of the child, you had to write barnet. Definitive articles are only added as suffixes in Norwegian (Bokmal), Swedish and Danish.